
 For \(\beta >1\) , \(W_\beta (z):=\prod _{j=0}^\infty (1-z/\beta ^j)\)  (\(z\in ) defines a nonzero entire function \) W\(, all of whose zeros are simple.\item  Let \) u=(uk)k=0\(. The function \) Fu\( below is entire and satisfies \) Fu(k)=uk\(, for all \) k0\(:\begin{equation}F_u^\beta (z):=\sum _{k=0}^\infty \frac{u_k}{W_\beta ^{\prime }(\beta ^k)}\cdot \frac{W_\beta (z)}{z-\beta ^k}.\end{equation}\item  When \) >2\(, we have the following quantitative estimate for \) l=1,2,3,...\(:\begin{equation}\sum _{k=1}^\infty \Big |\frac{W_\beta (\beta ^{-l})}{W_\beta ^{\prime }(\beta ^k)\cdot (\beta ^{-l}-\beta ^k)}\Big |\le \frac{\beta ^3\exp \frac{2}{\beta -1}}{(\beta ^2-1)(\beta ^2-\beta -1)}.\end{equation}\)
REF  is the result of the Weierstrass factorization theorem, see [1]} for example.
For , one can see that for \(k\ge 0\) ,
\(W_\beta ^{\prime }(\beta ^k)=\frac{W_\beta (z)}{z-\beta ^k}\Big |_{z=\beta ^k}=-\beta ^{-k}\prod _{j\ge 0;j\ne k}(1-\beta ^k/\beta ^j)=-\beta ^{-k}W_\beta (\beta ^{-1})\cdot \prod _{l=1}^{k}(1-\beta ^l).\) 
Since \(\beta ^l-1\ge \beta ^{l-1}(\beta -1)\)  for \(l\ge 1\) , we see that for \(k\ge 0\) :
\(|W_\beta ^{\prime }(\beta ^k)|\ge |W_\beta (\beta ^{-1})|\beta ^{-k}\prod _{l=1}^{k}\beta ^{l-1}(\beta -1)=|W_\beta (\beta ^{-1})|(\beta -1)^{k}\beta ^{-k}\cdot \beta ^{\frac{k^2-k}{2}}.\) 
Since \(\beta >1\) , we see that the sum \(\sum _{k=0}^\infty {u_k}/{W^{\prime }_\beta (\beta ^k)}\)  absolutely converges. Therefore the sum in () converges absolutely and locally uniformly in \(z\) . We conclude that \(F_u^\beta \)  is indeed an entire function.
For each \(j\ge 0\) , we have \(\frac{W_\beta (z)}{z-\beta ^k}\big |_{z=\beta ^j}=W^{\prime }_\beta (\beta ^k)\)  when \(j=k\)  and \(\frac{W_\beta (z)}{z-\beta ^k}\Big |_{z=\beta ^j}=0\)  when \(j\ne k\) . Therefore \(F_u^\beta (\beta ^k)=u_k\) , as for each \(z=\beta ^k\)  there is only one nonzero term in the sum .
For , note that for each \(l\ge 1\) , \(W_\beta (\beta ^{-l})=|W_\beta (\beta ^{-l})|<1\)  because \(|1-\beta ^{-l}/\beta ^j|<1\)  for each \(j\ge 0\) . Therefore using (REF ) and (REF ) we see that for \(k,l\ge 1\) ,
\(\Big |\frac{W_\beta (\beta ^{-l})}{W_\beta ^{\prime }(\beta ^k)(\beta ^{-l}-\beta ^k)}\Big |\le \frac{W_\beta (\beta ^{-1})^{-1}\cdot W_\beta (\beta ^{-l})\cdot \beta ^k}{(\beta -1)^{k}\beta ^{\frac{k-1}{2}k}\cdot |\beta ^{-l}-\beta ^k|}\le \frac{W_\beta (\beta ^{-1})^{-1}}{(\beta -1)^k\beta ^{k-1}}\cdot \frac{1}{1-\beta ^{-k-l}}\le \frac{W_\beta (\beta ^{-1})^{-1}}{(\beta ^2-\beta )^k}\cdot \frac{\beta }{1-\beta ^{-2}}.\) 
On the other hand
\(\big (W_\beta (\beta ^{-1})\big )^{-1}& =\prod _{j=1}^\infty (1-\beta ^{-j})^{-1}=\exp \sum _{j=1}^\infty \log \frac{1}{1-\beta ^{-j}}\\&<\exp \sum _{j=1}^\infty \log (1+2\beta ^{-j})&(\tfrac{1}{1-x}<1+2x\text{ when }0<x<\tfrac{1}{2})\\&<\exp \sum _{j=1}^\infty 2\beta ^{-j}=\exp \frac{2}{\beta -1}&(\log (1+x)<x\text{ when }0<x<1).\) 

Therefore taking sum over \(k\ge 1\)  we obtain ():
\(\sum _{k=1}^\infty \left|\frac{W_\beta (\beta ^{-l})}{W_\beta ^{\prime }(\beta ^k)(\beta ^{-l}-\beta ^k)}\right|\le \frac{\beta }{1-\beta ^{-2}}\sum _{k=1}^\infty \frac{\exp \frac{2}{\beta -1}}{(\beta ^2-\beta )^{k}}=\frac{\beta ^3\exp \frac{2}{\beta -1}}{(\beta ^2-1)(\beta ^2-\beta -1)}.\) 
Remark 12 
If one takes \(u_k:=(-1)^k\)  in Lemma REF  , then the corresponding sequence \(a_j:=\frac{1}{j!}(F_u^\beta )^{(j)}(0)\)  (\(j=0,1,2,\dots \) ) satisfies \(\sum _j a_j\beta ^{jk}=(-1)^k\)  for all \(k\ge 0\) . This is enough to prove the boundedness results in [2]} (\(C^k\)  for \(k\ge 0\) ), but in our case we also need the equality to be true for \(k<0\) . However, one cannot construct an entire function \(F\)  such that \(F(2^{-k})=(-1)^{-k}\)  for all \(k\ge 0\) .

Remark 13 
Alternatively, one can construct the \((a_j)_j\)  using Vandermonde determinant, which is the method of [2]}. Let \(M_1,M_2\in \mathbb {N}_0\)  and let \(b_{-M_1},\dots ,b_{M_2}>0\)  be distinct numbers. By Cramer's rule we see that the solutions of the equation system \(\sum _{j=-M_1}^{M_2}a_jb_j^k=(-1)^k\)  for \(-M_1\le k\le M_2\)  are (uniquely) determined by the following (see [4]} for example):
\(a_j=(-b_j)^{M_1}\frac{V(b_{-M_1},\dots ,b_{j-1},-1,b_{j+1},\dots ,b_{M_2})}{V(b_{-M_1},\dots ,b_{M_2})}=(-b_j)^{M_1}\hspace{-7.22743pt}\prod _{\begin{array}{c}-M_1\le k\le M_2\\k\ne j\end{array}}\hspace{-3.61371pt}\frac{b_k+1}{b_k-b_j},\ -M_1\le j\le M_2.\) 
Here \(V(x_{-M_1},\dots ,x_{M_2})=\prod _{-M_1\le u<v\le M_2}(x_v-x_u)\) .
Now we set \(b_j:=\beta ^j\)  for some \(\beta >1\) . If we let \(M_1=0\)  and let \(M_2\rightarrow +\infty \) , we see that (REF ) converges and produces the same sequence \((a_j)_{j=0}^\infty \)  to Remark REF .

We find \((a_j)_{j=-\infty }^\infty \)  through the following:
Proposition 14 
There are real numbers \((\tilde{a}_j)_{j=0}^\infty \)  such that \(\sum _{j=0}^\infty \tilde{a}_j(4^{jk}+4^{-jk})=(-1)^k\)  for all \(k=0,1,2,\dots \) , with all the summations converging absolutely.
Therefore the sequences \(a_j:=\tilde{a}_{|j|}\)  (\(j\ne 0\) ) \(a_0:=\frac{1}{2}\tilde{a}_0\)  and \(b_j:=4^j\)  (\(j\in \mathbb {Z}\) ) fulfill condition (REF ).

The idea is to construct an iterated sequence \(\tilde{a}^\nu =(\tilde{a}_0^\nu ,\tilde{a}_1^\nu ,\tilde{a}_2^\nu ,\dots )\)  such that
\(\sum _{j=0}^\infty \tilde{a}_j^\nu 4^{jk}=(-1)^k-\sum _{j=0}^\infty \tilde{a}_j^{\nu -1}4^{-jk},\quad \forall \, k\ge 0.\) 
The limit \(\lim _{\nu \rightarrow \infty }\tilde{a}_j^\nu \)  will be our \(\tilde{a}_j\) , provided that it is convergent. In practice, it is more convenient to prove the convergence of \((u^\nu )_{\nu =0}^\infty \subset \ell ^\infty \)  via the correspondence \(u^\nu _k:=\sum _{j=0}^\infty \tilde{a}_j^\nu 4^{jk}\)  and \(\tilde{a}_j^\nu =\frac{1}{j!}\frac{d^j}{dz^j}F_{u^\nu }^4(z)|_{z=0}\) , where \(F_{u^\nu }^4(z)\)  is the entire function in Lemma REF   with \(\beta =4\)  and \(u=u^\nu \) .
Let \(u^0=(u^0_k)_{k=0}^\infty \in \ell ^\infty \)  be \(u^0_0:=\frac{1}{2}\)  and \(u^0_k:=(-1)^k\)  for \(k\ge 1\) . For each \(\nu \ge 1\) , we let \(u^\nu _0:=\frac{1}{2}\)  and
\(u^\nu _k:=(-1)^k-\sum _{j=0}^\infty \Big (\tfrac{1}{j!}\frac{d^j}{dz^j}F_{u^{\nu -1}}^4(z)\Big |_{z=0}\Big )4^{-jk}=(-1)^k-F_{u^{\nu -1}}^4(4^{-k}),\quad \text{for } k\ge 1.\) 
By Lemma REF  , if \(u^{\nu -1}\in \ell ^\infty \)  then \(F_{u^{\nu -1}}(z)\)  is entire in \(|z|<1\) . Since \(u^0\in \ell ^\infty \) , by recursion we get \(\Vert u^\nu \Vert _{\ell ^\infty }\le 1+\Vert F_{u^{\nu -1}}\Vert _{L^\infty [0,1]}<\infty \)  for every \(\nu \ge 1\) . Thus we have a sequence of bounded sequences \((u^\nu )_{\nu =0}^\infty \subset \ell ^\infty \) .
Moreover since \(u^\nu _0\equiv \frac{1}{2}\) , by (),
\(\Vert u^{\nu +1}-u^{\nu }\Vert _{\ell ^\infty }&=\sup _{l\ge 1}|u^{\nu +1}_l-u^\nu _l|\le \sup _{l\ge 1}|F_{u^\nu }^4(4^{-l})-F_{u^{\nu -1}}^4(4^{-l})|\\&\le \sup _{l\ge 1}\sum _{k=1}^\infty \Big |\frac{(u^\nu _k-u^{\nu -1}_k)\cdot W_4(4^{-l})}{W_4^{\prime }(4^k)\cdot (4^{-l}-4^k)}\Big |\le \Vert u^\nu -u^{\nu -1}\Vert _{\ell ^\infty }\frac{4^3\exp \frac{2}{4-1}}{(4^2-1)(4^2-4-1)}=\frac{64 e^{\frac{2}{3}}}{165} \cdot \Vert u^\nu -u^{\nu -1}\Vert _{\ell ^\infty }.\) 

Here \(\frac{64 e^{2/3}}{165}\approx 0.755<1\) . We conclude that \((u^\nu )_{\nu =0}^\infty \subset \ell ^\infty \)  is a contraction sequence and must have a limit \(u^\infty \in \ell ^\infty \) . Clearly \(u^\infty _0=\frac{1}{2}\)  and \(u^\infty _k=(-1)^k-F_{u^\infty }^4(4^{-k})\)  for \(k\ge 1\) .
By Lemma REF  , the function \(F^4_{u^\infty }(z)\)  is entire. We define \(\tilde{a}_j\)  in the way that \(F_{u^\infty }^4(z)=\sum _{j=0}^\infty \tilde{a}_jz^j\) , thus
\(\sum _{j=0}^\infty \tilde{a}_j&=u^\infty _0=1-u^\infty _0=(-1)^0-\sum _{j=0}^\infty \tilde{a}_j,& k=0,\\\sum _{j=0}^\infty \tilde{a}_j4^{jk}&=u^\infty _k=(-1)^k-F_{u^\infty }^4(4^{-k})=(-1)^k-\sum _{j=0}^\infty \tilde{a}_j4^{-jk},&k\ge 1.\) 

Since \(F^4_{u^\infty }\)  is entire, the sum \(\sum _{j=0}^\infty \tilde{a}_j4^{jk}(=F^4_{u^\infty }(4^k))\)  always converge absolutely. Therefore \((\tilde{a}_j)_{j=0}^\infty \)  is as desired.
Let \(a_j:=\tilde{a}_{|j|}\)  for \(j\ne 0\)  and \(a_0:=\frac{1}{2}\tilde{a}_0\) , we see that \(\sum _{j=-\infty }^\infty a_j4^{jk}=(-1)^k\)  for all \(k\in \mathbb {Z}\)  with all summations converging absolutely. In particular \(\sum _{j=-\infty }^\infty |a_j|4^{|j|k}<\infty \)  for all \(k\ge 1\) , which means that \(\sum _{j=-\infty }^\infty |a_j|2^{\delta |j|}4^{jk}<\infty \)  for all \(\delta >0\)  and \(k\in \mathbb {Z}\) . Therefore \((a_j,4^j)_{j=-\infty }^\infty \)  satisfies the condition (REF ) and we prove Theorem REF  REF .
The direct computations yield the following. We omit the proof to reader.
Lemma 15

 Let \((a,b)=(a_j,b_j)_{j=-\infty }^\infty \)  be the collection satisfying the condition (REF ). Then for every \(k\in \mathbb {Z}\)  the collections \((a(-b)^k,b):=(a_j(-b_j)^k,b_j)_{j=-\infty }^\infty \)  and \((a(-b)^k,\frac{1}{b}):=(a_{j}(-b_{j})^k,\frac{1}{b_{j}})_{j=-\infty }^\infty \)  also satisfy (REF ).

 For multi-index \(\gamma \in \mathbb {N}_0^n\)  we have \(\partial ^\gamma \circ E^{a,b}=E^{a(-b)^{\gamma _n},b}\circ \partial ^\gamma \)  and \(E^{a,b}\circ \partial ^\gamma =\partial ^\gamma \circ E^{a(-b)^{-\gamma _n},b}\) .

 Let \(E=E^{a,b}\)  be the extension operator defined in (REF ). Then
\(E^*g(x^{\prime },x_n)=g(x^{\prime },x_n)-\sum _{j=-\infty }^\infty \frac{a_{j}}{-b_{j}}g\big (x^{\prime },-\tfrac{1}{b_{j}}\cdot x_n\big ),\quad x^{\prime }\in \mathbb {R}^{n-1},\quad x_n>0.\) 


Remark 16 
In the notations of (REF ) and (REF ) we can write \(E^{a,b}=S+\sum _ja_j\vartheta ^{-b_j}\circ S\) . For \(r<0\)  we have
\(\partial _{x_n}(\vartheta ^rSf)(x)=r(S\partial _{x_n}f)(x^{\prime },rx_n)-\delta _0(x_n)f(x^{\prime },rx_n),\) 
where \(\delta _0\)  is the Dirac measure at \(0\in \mathbb {R}\) .
Therefore \(\partial _n\circ E^{a,b}=E^{a(-b),b}\circ \partial _n\)  (in the domain where either side is defined) if and only if \(\sum _ja_j=1\) .
Taking higher order derivatives we see that \(\partial ^\gamma \circ E^{a,b}=E^{a(-b)^{\gamma _n},b}\circ \partial ^\gamma \)  still holds if we only assume the condition \(\sum _ja_j(-b_j)^k=1\)  for \(0\le k\le \gamma _n\) .

Lemma 17 
Let \((X,\Vert \cdot \Vert )\)  be a quasi-Banach space such that \((x,y)\mapsto \Vert x-y\Vert ^q\)  is a metric for some \(0<q\le 1\) , i.e. \(\Vert x+y\Vert ^q\le \Vert x\Vert ^q+\Vert y\Vert ^q\)  for all \(x,y\in X\) . Then for any \(\delta >0\)  there is a constant \(K_{q,\delta }>0\)  such that
\(\big \Vert \sum _{j=-\infty }^\infty x_j\big \Vert \le K_{q,\delta }\sum _{j=-\infty }^\infty 2^{\delta |j|}\Vert x_j\Vert \) 
for all sequence \((x_j)_{j\in \mathbb {Z}}\subset X\)  such that right hand summation converges. In this case the sum \(\sum _{j=-\infty }^\infty x_j\)  converges with respect to the quasi-norm topology of \(X\) .

Remark 18 The couple \((X,\Vert \cdot \Vert ^q)\)  is also call a \(q\) -Banach space.
In practice we use \(X=W^{k,q}(U)\)  and \(X={A}_{pr}^{s\tau }(U)\)  (see Definition REF ) with \(q=\min (p,r,1)\) .

When \((x_j)_{j\in \mathbb {Z}}\)  are finitely nonzero, by Hölder's inequality we have
\(\Big \Vert  {\sum _j x_j} \Big \Vert _X\le \Big |\sum _j\Vert x_j\Vert _X^q\Big |^{1/q}\le \left(\big \Vert _{\ell ^{\frac{1}{q}}(\mathbb {Z})}\big \Vert (2^{-\delta q|j|})_{j\in \mathbb {Z}} \big \Vert (2^{\delta q|j|}\Vert x_j\Vert _X^q)_{j\in \mathbb {Z}}\big \Vert _{\ell ^{\frac{1}{1-q}}(\mathbb {Z})}\right)^{1/q}=K_{q,\delta } \sum _j 2^{\delta |j|}\Vert {x_j} \Vert _X. \) 
Here \(K_{q,\delta }:=(\frac{2^{\delta /(1-q)}+1}{2^{\delta /(1-q)}-1})^{1/q-1}\)  if \(q<1\)  and \(K_{q,\delta }:=1\)  if \(q=1\) .
When \(\sum _{j=-\infty }^\infty 2^{\delta |j|}\Vert x_j\Vert <\infty \) , we have \(\lim \limits _{\nu \rightarrow +\infty }\sup \limits _{\nu \le \nu ^{\prime }<\nu ^{\prime \prime }}\big \Vert \sum \limits _{\nu ^{\prime }<|j|\le \nu ^{\prime \prime }}x_j\big \Vert \le K_{q,\delta }\lim \limits _{\nu \rightarrow +\infty }\sum \limits _{|j|\ge \nu }2^{\delta |j|}\Vert x_j\Vert =0\) . Therefore \(\big (\sum _{|j|\le \nu }x_j\big )_{\nu =1}^\infty \)  is a Cauchy sequence in \(X\) . Since \(X\)  is complete, the sum must converge to a unique limit element and the same inequality \(\Vert \sum _jx_j\Vert \le K_{q,\delta }\sum _j2^{\delta |j|}\Vert x_j\Vert \)  holds.
The proof of boundedness on Sobolev spaces is standard:
We only need to prove the Sobolev boundedness \(W^{k,p}\)  for \(k\ge 0\) , \(0<p\le \infty \) .
Indeed suppose we get \(E:W^{k,\infty }(\mathbb {R}^n_+)\rightarrow W^{k,\infty }(\mathbb {R}^n)\) , then \(C^\infty (U)=\bigcap _{k=0}^\infty W^{k,\infty }(U)\)  for \(U\in \lbrace \mathbb {R}^n,\mathbb {R}^n_+\rbrace \)  gives \(E:C^\infty (\mathbb {R}^n_+)\rightarrow C^\infty (\mathbb {R}^n)\) . Clearly \(C^k(U)=\overline{C^\infty (U)}^{\Vert \cdot \Vert _{W^{k,\infty }(U)}}\)  for \(U\in \lbrace \mathbb {R}^n,\mathbb {R}^n_+\rbrace \) , thus \(E:C^k(\mathbb {R}^n_+)\rightarrow C^k(\mathbb {R}^n)\)  holds for all \(k\ge 0\) .
The \(C^{k,s}\)  boundedness (\(k\in \mathbb {Z}\) , \(0<s<1\) ) can follow either from the same argument for \(W^{k,p}\)  below, or the Besov correspondence \(C^{k,s}(U)={B}_{\infty \infty }^{k+s}(U)\) , for \(U\in \lbrace \mathbb {R}^n,\mathbb {R}^n_+\rbrace \)  (see Proposition REF  REF ).
When \(k\ge 0\)  one can also obtain the boundedness by the interpolation \(C^{k,s}(U)=(C^k(U),C^{k+1}(U))_{s,\infty }\)  (see for example [5]} and the proof of [5]}).
For the \(W^{k,p}\) -boundedness, by Lemma REF , for every \(f\in L^p(\mathbb {R}^n_+)\)  if \(1\le p\le \infty \) , and every \(f\in C_c^\infty (\overline{\mathbb {R}^n_+})\)  if \(0<p<1\) , we have (recall the notation \(\vartheta ^r\)  in (REF ))
\(\begin{aligned}\Vert E^{a,b}f\Vert _{L^p(\mathbb {R}^n)}\le &\textstyle 2^{\max (0,\frac{1}{p}-1)}\big (\Vert  f\Vert _{L^p(\mathbb {R}^n_+)}+\big \Vert \sum _{j\in \mathbb {Z}}a_j\cdot \vartheta ^{-b_j}f\big \Vert _{L^p(\mathbb {R}^n_-)}\big )\\\lesssim &_{p,\delta }\Vert  f\Vert _{L^p(\mathbb {R}^n_+)}+\sum _{j\in \mathbb {Z}}2^{\delta |j|}|a_j|\Big (\int _{\mathbb {R}^n_-}|f(x^{\prime },-b_jx_n)|^pdx^{\prime }dx_n\Big )^\frac{1}{p}\\=&\Vert f\Vert _{L^p(\mathbb {R}^n_+)}+\sum _{j\in \mathbb {Z}}2^{\delta |j|}|a_j|\Big (b_j^{-1}\int _{\mathbb {R}^n_-}|f(x^{\prime },y_n)|^pdx^{\prime }dy_n\Big )^\frac{1}{p}\\=&\Big (1+\sum _{j\in \mathbb {Z}}2^{\delta |j|}|a_j|b_j^{-\frac{1}{p}}\Big )\Vert f\Vert _{L^p(\mathbb {R}^n_+)}.\end{aligned}\) 
When \(k\ge 0\) , also for every \(f\in L^p(\mathbb {R}^n_+)\)  if \(1\le p\le \infty \) , and every \(f\in C_c^\infty (\overline{\mathbb {R}^n_+})\)  if \(0<p<1\) , applying Lemma REF  REF  to (REF ) we have
\(\Vert E^{a,b}f\Vert _{W^{k,p}(\mathbb {R}^n)}\lesssim &_{k,p}\sum _{|\gamma |\le k}\Vert \partial ^\gamma E^{a,b} f\Vert _{L^p(\mathbb {R}^n)}=\sum _{|\gamma |\le k}\Vert E^{a(-b)^{\gamma _n},b}\partial ^\gamma f\Vert _{L^p(\mathbb {R}^n)}\\\lesssim &_{p,\delta }\sum _{|\gamma |\le k}\sum _{j\in \mathbb {Z}}2^{\delta |j|}|a_j|b_j^{\gamma _n-\frac{1}{p}}\Vert \partial ^\gamma f\Vert _{L^p(\mathbb {R}^n_+)}\lesssim _{k,p}\sum _{j\in \mathbb {Z}}2^{\delta |j|}|a_j|(b_j^{-\frac{1}{p}}+b_j^{k-\frac{1}{p}})\Vert  f\Vert _{W^{k,p}(\mathbb {R}^n_+)}.\) 
By the assumption (REF ) we can take \(\delta >0\)  small such that \(\sum _{j\in \mathbb {Z}}2^{\delta |j|}|a_j|b_j^{k-\lceil \frac{1}{p}\rceil }<\infty \)  and \(\sum _{j\in \mathbb {Z}}2^{\delta |j|}|a_j|b_j^{k}<\infty \) . We now get the boundedness of \(W^{k,p}\)  for \(k\ge 0\) .
When \(k<0\) , applying Lemma REF  REF  to (REF ) for

every \(f\in W^{k,p}(\mathbb {R}^n_+)\)  and \(\lbrace g_\gamma \rbrace _{|\gamma |\le -k}\subset L^p(\mathbb {R}^n_+)\)  such that \(f=\sum _{|\gamma |\le -k}\partial ^\gamma g_\gamma \) , if \(1\le p\le \infty \) ;

every \(f\in C_c^\infty (\overline{\mathbb {R}^n_+})\)  and \(\lbrace g_\gamma \rbrace _{|\gamma |\le -k}\subset C_c^\infty (\overline{\mathbb {R}^n_+})\)  such that \(f=\sum _{|\gamma |\le -k}\partial ^\gamma g_\gamma \) , if \(0<p<1\) ,

we have the following
\(\Vert E^{a,b}f\Vert _{W^{k,p}(\mathbb {R}^n)}\lesssim &_{k,p}\sum _{|\gamma |\le -k}\Vert  E^{a,b}\partial ^\gamma g_\gamma \Vert _{W^{k,p}(\mathbb {R}^n)}=\sum _{|\gamma |\le -k}\Vert \partial ^\gamma E^{a(-b)^{-\gamma _n},b} g_\gamma \Vert _{W^{k,p}(\mathbb {R}^n)}\\\le &\sum _{|\gamma |\le -k}\Vert E^{a(-b)^{-\gamma _n},b} g_\gamma \Vert _{L^p(\mathbb {R}^n)}\lesssim _{p,\delta }\sum _{|\gamma |\le -k}\sum _{j\in \mathbb {Z}}2^{\delta |j|}|a_j|b_j^{-\gamma _n-\frac{1}{p}}\Vert g_\gamma \Vert _{L^p(\mathbb {R}^n_+)}\\\le &\sum _{j\in \mathbb {Z}}2^{\delta |j|}|a_j|(b_j^{-\frac{1}{p}}+b_j^{k-\frac{1}{p}})\sum _{|\gamma |\le -k}\Vert g_\gamma \Vert _{L^p(\mathbb {R}^n_+)}.\) 
Taking the infimum over all \(\lbrace g_\gamma \rbrace _{|\gamma |\le -k}\)  for \(f\) , and taking \(\delta \)  from (REF ) such that \(\sum _{j\in \mathbb {Z}}2^{\delta |j|}|a_j|(b_j^k+b_j^{k-\lceil \frac{1}{p}\rceil })<\infty \)  as well, we get the \(W^{k,p}\) -boundedness for \(k<0\) , finishing the proof.
Remark 19 
The result can be extended to vector-valued functions. Indeed let \(0<q\le 1\) , \(0<p<\infty \) , and let \((X,|\cdot |_X)\)  be a quasi Banach space such that \(|x+y|_X^q\le |x|_X^q+|y|_X^q\) . Then for every \(L^p\)  (strongly measurable) function \(f:\mathbb {R}^n_+\rightarrow X\) , by the same calculation to (REF ),
\(\begin{aligned}&\bigg (\int _{\mathbb {R}^n_-}\Big |\sum _{j\in \mathbb {Z}}a_jf(x^{\prime },-b_jx_n)\Big |_X^pdx^{\prime }dx_n\bigg )^\frac{1}{p}\le K_{q,\frac{\delta }{2}}\bigg (\int _{\mathbb {R}^n_-}\Big (\sum _{j\in \mathbb {Z}}2^{|j|\frac{\delta }{2}}|a_j||f(x^{\prime },-b_jx_n)|_X\Big )^pdx^{\prime }dx_n\bigg )^\frac{1}{p}\\\le &K_{q,\frac{\delta }{2}}K_{p,\frac{\delta }{2}}\sum _{j\in \mathbb {Z}}2^{|j|(\frac{\delta }{2}+\frac{\delta }{2})}|a_j|\bigg (\int _{\mathbb {R}^n_+}b_j^{-1}|f(x^{\prime },y_n)|_X^pdx^{\prime }dy_n\bigg )^\frac{1}{p}=\Big (K_{q,\frac{\delta }{2}}K_{p,\frac{\delta }{2}}\sum _{j\in \mathbb {Z}}2^{|j|\delta }|a_j|b_j^{-\frac{1}{p}}\Big )\Vert f\Vert _{L^p(\mathbb {R}^n_+;X)}.\end{aligned}\) 
Here \(K_{q,\delta /2}\)  and \(K_{p,\delta /2}\)  are the constants in Lemma REF .
Also by Lemma REF  the sum of functions \(\sum _{j\in \mathbb {Z}}a_jf(x^{\prime },-b_jx_n)\)  converges in \(L^p(\mathbb {R}^n_-;X)\) .
Therefore \(E^{a,b}\)  defines a bounded linear map \(E^{a,b}:L^p(\mathbb {R}^n_+;X)\rightarrow L^p(\mathbb {R}^n;X)\)  with the control of operator norms \(\Vert E^{a,b}\Vert _{L^p(\mathbb {R}^n_+;X)\rightarrow L^p(\mathbb {R}^n;X)}\lesssim _{p,q,\delta }\sum _{j\in \mathbb {Z}}2^{|j|\delta }|a_j|b_j^{-1/p}\) .
In practice we will take \(X={B}_{pq}^s(\mathbb {R}^{n-1})\)  in the proof of Theorem REF , Step A3.

Boundedness on Besov and Triebel-Lizorkin: Proof of Theorem REF  REF  and  REF
Before we discuss the boundedness on Besov and Triebel-Lizorkin spaces, for completeness we prove that \(E:{S}^{\prime }\rightarrow {S}^{\prime }\)  is defined. Recall Definition REF  that \({S}^{\prime }(\mathbb {R}^n_+)={S}^{\prime }(\mathbb {R}^n)/\lbrace \tilde{f}:\tilde{f}|_{\mathbb {R}^n_+}=0\rbrace \) .
Lemma 20 \(E:{S}^{\prime }(\mathbb {R}^n_+)\rightarrow {S}^{\prime }(\mathbb {R}^n)\)  is defined and bounded.

Since \({S}^{\prime }(\mathbb {R}^n_+)={S}(\mathbb {R}^n_+)^{\prime }\)  (see Remark REF ), by duality it is equivalent to show that \(E^*:{S}(\mathbb {R}^n)\rightarrow {S}(\mathbb {R}^n_+)\)  is continuous.
Let \(g\in {S}(\mathbb {R}^n)\) , using (REF ) we see that for every \(\alpha ,\beta \in \mathbb {N}_0^n\)  and \(x\in \mathbb {R}^n_+\) ,
\(|x^\alpha \partial ^\beta E^*g(x)| \le |x^\alpha \partial ^\beta g(x)|+\sum _{j\in \mathbb {Z}}\big |a_j(-b_j)^{\alpha _n-\beta _n-1}\big (x^{\prime },-\tfrac{1}{b_j}\cdot x_n\big )^\alpha (\partial ^\beta g) (x^{\prime },-\tfrac{x_n}{b_j})\big |.\) 
By (REF ), \(\sum _{j\in \mathbb {Z}}|a_j|\cdot b_j^{\alpha _n-\beta _n-1}<\infty \) . We conclude that \(|x^\alpha \partial ^\beta E^*g(x)|\lesssim _{a,b,\alpha ,\beta }\sup _{y\in \mathbb {R}^n}|y^\alpha \partial ^\beta g(y)|\)  for all \(x\in \mathbb {R}^n_+\) .
On the other hand, by the condition \(\sum _ja_j(-b_j)^{-k}=1\)  for \(k=1,2,3,\dots \)  in (REF ), we have
\(\lim _{x_n\rightarrow 0+}\partial ^\beta E^*g(x)=\partial ^\beta g(x^{\prime },0)-\sum _{j\in \mathbb {Z}}a_j(-b_j)^{-\beta _n-1}(\partial ^\beta g)(x^{\prime },0)=0,\quad \forall \beta \in \mathbb {N}_0^n.\) 
Therefore along with the zero extension to \(\mathbb {R}^n_-\) , \(E^*g\)  defines a Schwartz function in \(\mathbb {R}^n\)  which has support \(\overline{\mathbb {R}^n_+}\) . Moreover \(\sup _{x\in \mathbb {R}^n}|x^\alpha \partial ^\beta E^*g(x)|\lesssim _{a,b,\alpha ,\beta }\sup _{y\in \mathbb {R}^n}|y^\alpha \partial ^\beta g(y)|\)  with the implied constant being independent of \(g\) .
Since the seminorms \(\big \lbrace \sup _{x\in \mathbb {R}^n}|x^\alpha \partial ^\beta E^*g(x)|\big \rbrace _{\alpha ,\beta \in \mathbb {N}_0^n}\)  define the topology of \({S}(\mathbb {R}^n)\) , we conclude that
\(E^*:{S}(\mathbb {R}^n)\rightarrow {S}(\mathbb {R}^n_+)\)  is continuous. Therefore \(E:{S}^{\prime }(\mathbb {R}^n_+)\rightarrow {S}^{\prime }(\mathbb {R}^n)\)  is continuous as well.
We now turn to prove Theorem REF  REF  and REF .
To prove Theorem REF  REF , it might be possible to use the mean oscillation approach in [7]}. However [7]} fails when \((a_j)_j\)  has infinite nonzero terms. This is critical if one wants to prove the Besov and Triebel-Lizorkin boundedness of Seeley's operator (see Remark REF ) using such method. The authors do not know what the correct modification of [7]} is.
Instead, we adapt the arguments from [10]}. The proof involves characteristic function multiplier [10]}, the Fubini decomposition [10]} and interpolations. We recall these facts from literature.
Proposition 21 
Let \(0<p,q\le \infty \) , \(s\in \mathbb {R}\)  and \(0\le \tau \le \frac{1}{p}\) . Let \(U\subseteq \mathbb {R}^n\)  be an arbitrary open subset.

 For \({A}\in \lbrace {B},{F}\rbrace \)  and for every integer \(m\ge 1\) , \({A}_{pq}^s(U)=\big \lbrace \sum _{|\alpha |\le m}\partial ^\alpha g_\alpha :g_\alpha \in {A}_{pq}^{s+m}(U)\big \rbrace \)  and has equivalent (quasi-)norm \(\Vert f\Vert _{{A}_{pq}^s(U)}\approx _{p,q,s,m}\inf \big \lbrace \sum \limits _{|\alpha |\le m}\Vert g_\alpha \Vert _{{A}_{pq}^{s+m}(U)}:f=\sum \limits _{|\alpha |\le m}\partial ^\alpha g_\alpha \text{ as distributions}\big \rbrace \) .

If \(U\in \lbrace \mathbb {R}^n,\mathbb {R}^n_+\rbrace \)  or \(U\)  is a bounded smooth domain, then \(C^{k,s}(U)={B}_{\infty \infty }^s(U)\)  for all \(k\in \mathbb {Z}\)  and \(0<s<1\) , with equivalent norms.

(Interpolations)  For \(s_0,s_1\in \mathbb {R}\)  and \(\theta \in (0,1)\)  such that \(s_0\ne s_1\)  and \(s=(1-\theta )s_0+\theta s_1\) , we have
\(\text{\normalfont complex interpolation}:&\quad [{F}_{pq}^{s_0}(\mathbb {R}^n),{F}_{pq}^{s_1}(\mathbb {R}^n)]_\theta ={F}_{pq}^s(\mathbb {R}^n);\\\text{\normalfont real interpolation}:&\quad ({F}_{pp}^{s_0}(\mathbb {R}^n),{F}_{pp}^{s_1}(\mathbb {R}^n))_{\theta ,q}={B}_{pq}^{s}(\mathbb {R}^n);\\&\quad ({F}_{pp}^{s_0,\tau }(\mathbb {R}^n),{F}_{pp}^{s_1,\tau }(\mathbb {R}^n))_{\theta ,q}={N}_{pq}^{s\tau }(\mathbb {R}^n),& \text{for }\tau \in [0,\tfrac{1}{p})\quad (\text{provided }p<\infty ).\) 

(Fubini) When \(p<\infty \)  or \(p=q=\infty \) , and when \(s>n\max (0,\frac{1}{p}-1,\frac{1}{q}-1)\) , we have decomposition \({F}_{pq}^s(\mathbb {R}^n)=L^p(\mathbb {R}_{x_n};{F}_{pq}^s(\mathbb {R}^{n-1}_{x^{\prime }}))\cap L^p(\mathbb {R}^{n-1}_{x^{\prime }};{F}_{pq}^s(\mathbb {R}_{x_n}))\) , with equivalent (quasi-)norms.

(Local representations) For \(s<0\) , we have equivalent norms
\(\Vert f\Vert _{{B}_{pq}^{s\tau }(\mathbb {R}^n)}\approx &_{n,p,q,s,\tau }\sup _{x\in \mathbb {R}^n;R>0}R^{-n\tau }\Vert f\Vert _{{B}_{pq}^s(B(x,R))};\\\Vert f\Vert _{{F}_{pq}^{s\tau }(\mathbb {R}^n)}\approx &_{n,p,q,s,\tau }\sup _{x\in \mathbb {R}^n;R>0}R^{-n\tau }\Vert f\Vert _{{F}_{pq}^s(B(x,R))},\qquad \text{provided }p<\infty .\) 


For REF , by [13]} \(\Vert \partial ^\alpha \tilde{g}\Vert _{{A}_{pq}^s(\mathbb {R}^n)}\lesssim \Vert \tilde{g}\Vert _{{A}_{pq}^{s+m}(\mathbb {R}^n)}\)  holds for \(|\alpha |\le m\) . Note that \(\tilde{g}|_U=g\)  implies \(\partial ^\alpha \tilde{g}|_U=\partial ^\alpha g\) , thus taking restrictions we get \(\Vert \sum _{|\alpha |\le m}\partial ^\alpha g_\alpha \Vert _{{A}_{pq}^s(U)}\lesssim \sum _{|\alpha |\le m}\Vert g_\alpha \Vert _{{A}_{pq}^{s+m}(U)}\) . Therefore \(\Vert f\Vert _{{A}_{pq}^s(U)}\lesssim \sum _{|\alpha |\le m}\Vert g_\alpha \Vert _{{A}_{pq}^{s+m}(U)}\)  whenever \(f=\sum _{|\alpha |\le m}\partial ^\alpha g_\alpha \) .
Conversely for \(f\in {A}_{pq}^s(U)\) , take an extension \(\tilde{f}\in {A}_{pq}^s(\mathbb {R}^n)\) , we have \(\tilde{f}=(I-\Delta )(I-\Delta )^{-1}\tilde{f}=\tilde{g}_0+\sum _{j=1}^n\partial _j\tilde{g}_j\)  where \(\tilde{g}_0:=(I-\Delta )^{-1}\tilde{f}\)  and \(\tilde{g}_j:=-\partial _j(I-\Delta )^{-1}\tilde{f}\)  (\(1\le j\le n\) ). Thus by [13]} we have \(\sum _{j=0}^n\Vert \tilde{g}_j\Vert _{{A}_{pq}^{s+1}(\mathbb {R}^n)}\lesssim \Vert \tilde{f}\Vert _{{A}_{pq}^s(\mathbb {R}^n)}\) . Taking infimum of all extensions \(\tilde{f}\)  over \(f\)  we get REF  for \(m=1\) . Doing this recursively we get all \(m\ge 1\) .
For REF , we have \(C^{k,s}(\mathbb {R}^n)={B}_{\infty \infty }^{k+s}(\mathbb {R}^n)\)  from [10]} for \(k\ge 0\)  and \(0<s<1\) . Thus in this case \({B}_{\infty \infty }^{k+s}(U)=\lbrace \tilde{f}|_U:\tilde{f}\in C^{k,s}(\mathbb {R}^n)\rbrace \subseteq C^{k,s}(U)\)  for arbitrary open subset \(U\) . When \(U=\mathbb {R}^n_+\)  or \(U\)  is bounded smooth, by the existence of extension operator \(C^{k,s}(U)\rightarrow C^{k,s}(\mathbb {R}^n)\)  (see [16]}) we get \(C^{k,s}(U)\subseteq {B}_{\infty \infty }^{k+s}(U)\) . This proves the case \(k\ge 0\) .
Comparing the result REF  and (REF ) we see that \(C^{k,s}(U)={B}_{\infty \infty }^{k+s}(U)\)  for \(k<0\)  as well.
See [17]} and [18]} for REF .
See [13]} and [20]} for REF .
See [21]} for REF . In the reference the collection \(\lbrace B(x,2^{-J}):x\in \mathbb {R}^n,J\in \mathbb {Z}\rbrace \)  is replaced by \(\lbrace 2Q_{J,v}:=2^{-J}(v+(-\frac{1}{2},\frac{3}{2})^n):J\in \mathbb {Z},v\in \mathbb {Z}^n\rbrace \) , but there is no difference between the proofs. Note that one cannot use \(\lbrace Q_{J,v}:=2^{-J}(v+(0,1)^n):J\in \mathbb {Z},v\in \mathbb {Z}^n\rbrace \)  because for \(J_0\in \mathbb {Z}\) , the set \(\lbrace x\in \mathbb {R}^n:x_k\in 2^{-J_0}\mathbb {Z},\ \exists \,1\le k\le n\rbrace \)  is not contained in the union \(\bigcup _{J\ge J_0;v\in \mathbb {Z}^n}Q_{J,v}\) .
Recall the dilation operator \((\vartheta ^rf)(x):=f(x^{\prime },rx_n)\)  for \(r\in \mathbb {R}\backslash \lbrace 0\rbrace \)  from (REF ).
Proposition 22 For any \(0<p,q\le \infty \)  and \(s\in \mathbb {R}\)  there is a \(M=M_{p,q}^{n,s}>0\)  and \(C=C_{n,p,q,s,M}>0\)  such that
\(\Vert \vartheta ^rf\Vert _{{A}_{pq}^s(\mathbb {R}^n)}\le C(|r|^{-M}+|r|^M)\Vert f\Vert _{{A}_{pq}^s(\mathbb {R}^n)},\quad \text{for }r\ne 0,\quad {A}\in \lbrace {B},{F}\rbrace ,\quad f\in {A}_{pq}^s(\mathbb {R}^n).\) 

We only need to prove the Triebel-Lizorkin case, i.e. \({A}={F}\) . Indeed suppose we get the case \({A}={F}\) , then by real interpolation (), for each \(0<p,q\le \infty \)  and \(s\in \mathbb {R}\)  there is a \(C_{p,q,s}>0\)  such that (see [5]}, in fact \(C_{p,q,s}=C_q\) )
\(\Vert T\Vert _{{B}_{pq}^s(\mathbb {R}^n)\rightarrow {B}_{pq}^s(\mathbb {R}^n)}\le C_{p,q,s}(\Vert T\Vert _{{F}_{pp}^{s-1}(\mathbb {R}^n)\rightarrow {F}_{pp}^{s-1}(\mathbb {R}^n)}+\Vert T\Vert _{{F}_{pp}^{s+1}(\mathbb {R}^n)\rightarrow {F}_{pp}^{s+1}(\mathbb {R}^n)}),\) 
whenever \(T:{F}_{pp}^{s-1}(\mathbb {R}^n)\rightarrow {F}_{pp}^{s-1}(\mathbb {R}^n)\)  is a bounded linear map such that \(T:{F}_{pp}^{s+1}(\mathbb {R}^n)\rightarrow {F}_{pp}^{s+1}(\mathbb {R}^n)\)  is also bounded. Taking \(T=\vartheta ^r\)  and replacing \(M_{p,q}^{n,s}\)  (obtained from the \({F}\) -cases) with \(\max (M_{p,q}^{n,s},M_{p,p}^{n,s-1},M_{p,p}^{n,s+1})\)  we complete the proof.
Step 1: The case \(n=1\) , \(0<p<\infty \) , \(0<q\le \infty \)  and \(s>\frac{1}{p}-1\) . We use [23]}.
Recall \(\lambda =(\lambda _j)_{j=0}^\infty \)  from Definition REF . When \(s>\max (0,1-\frac{1}{p})\) , by [7]}, we have \(\Vert f\Vert _{{F}_{pq}^s(\mathbb {R})}\approx _{p,q,s}\Vert f\Vert _{L^p(\mathbb {R}^n)}+\big \Vert x\mapsto \big (\int _0^\infty t|(\lambda _1(\frac{\cdot }{t})\ast f)(x)|\frac{dt}{t^{sq+1}}\big )^{1/q}\big \Vert _{L^p(\mathbb {R})}\) . The same dilation argument from [23]} shows that
\(\Vert \vartheta ^rf\Vert _{{F}_{pq}^s(\mathbb {R})}\approx |r|^{-\frac{1}{p}}\Vert f\Vert _{L^p}+|r|^{s-\frac{1}{p}}\Big \Vert x\mapsto \Big (\int _0^\infty |(\lambda _1(\tfrac{\cdot }{t})\ast f)(x)|\frac{dt}{t^{sq}}\Big )^\frac{1}{q}\Big \Vert _{L^p}\lesssim _{p,q,s}(|r|^{-\frac{1}{p}}+|r|^{s-\frac{1}{p}})\Vert f\Vert _{{F}_{pq}^s(\mathbb {R})}.\) 
In particular we can take \(M_{p,q}^{1,s}:=\max (\frac{1}{p},s-\frac{1}{p},\frac{1}{p}-s)\)  in this case.
Step 2: The case \(n\ge 2\) , \(0<p<\infty \) , \(0<q\le \infty \)  and \(s>n\max (0,\frac{1}{p}-1,\frac{1}{q}-1)\) . We use Proposition REF  REF :
\(\Vert \vartheta ^rf\Vert _{{F}_{pq}^s(\mathbb {R}^n)}\approx &\Big (\int _{\mathbb {R}^{n-1}}\Vert \vartheta ^r(f(x^{\prime },\cdot ))\Vert _{{F}_{pq}^s(\mathbb {R})}^pdx^{\prime }\Big )^{1/p}+\Big (\int _{\mathbb {R}}\Vert f(\cdot ,rx_n)\Vert _{{F}_{pq}^s(\mathbb {R}^{n-1})}^pdx_n\Big )^{1/p}\\\lesssim &(|r|^{-\frac{1}{p}}+|r|^{s-\frac{1}{p}})\Big (\int _{\mathbb {R}^{n-1}}\Vert f(x^{\prime },\cdot )\Vert _{{F}_{pq}^s(\mathbb {R})}^pdx^{\prime }\Big )^{\frac{1}{p}}+|r|^{-\frac{1}{p}}\Big (\int _{\mathbb {R}}\Vert f(\cdot ,y_n)\Vert _{{F}_{pq}^s(\mathbb {R}^{n-1})}^pdy_n\Big )^{\frac{1}{p}}\\\lesssim &(|r|^{-M_{p,q}^{1,s}}+|r|^{M_{p,q}^{1,s}})\Vert f\Vert _{{F}_{pq}^s(\mathbb {R}^n)}.\) 

In particular we can take \(M_{p,q}^{n,s}:=M_{p,q}^{1,s}\)  in this case.
Step 3: The case \(0<p<\infty \) , \(0<q\le \infty \)  and \(s\in \mathbb {R}\) . We use Proposition REF  REF .
Let \(m\ge 0\)  be such that \(s+m>n\max (0,\frac{1}{p}-1,\frac{1}{q}-1)\) . Note that \(\vartheta ^\frac{1}{r}\circ \partial ^\alpha =r^{\alpha _n}\partial ^\alpha \circ \vartheta ^\frac{1}{r}\) , thus
\(&\Vert \vartheta ^rf\Vert _{{F}_{pq}^s(\mathbb {R}^n)}\approx \textstyle \inf \Big \lbrace \sum _{|\alpha |\le m}\Vert g_\alpha \Vert _{{F}_{pq}^{s+m}(\mathbb {R}^n)}:\vartheta ^rf=\sum _{|\alpha |\le m}\partial ^\alpha g_\alpha \Big \rbrace \\=&\inf \Big \lbrace \sum _{|\alpha |\le m}\Vert g_\alpha \Vert _{{F}_{pq}^{s+m}(\mathbb {R}^n)}:f=\sum _{|\alpha |\le m}r^{\alpha _n}\partial ^\alpha (\vartheta ^\frac{1}{r}g_\alpha )\Big \rbrace =\inf \Big \lbrace \sum _{|\alpha |\le m}\frac{\Vert \vartheta ^rh_\alpha \Vert _{{F}_{pq}^{s+m}(\mathbb {R}^n)}}{|r|^{\alpha _n}}:f=\sum _{|\alpha |\le m}\partial ^\alpha h_\alpha \Big \rbrace \\\lesssim &(|r|^{-m}+1)(|r|^{-M_{p,q}^{n,s+m}}+|r|^{M_{p,q}^{n,s+m}})\Vert f\Vert _{{F}_{pq}^s(\mathbb {R}^n)}.\) 

In particular we can take \(M_{p,q}^{n,s}:=M_{p,q}^{n,s+m}+m\)  in this case.
Step 4: The case \(p=\infty \) , \(0<q\le \infty \)  and \(s\in \mathbb {R}\) . We use Proposition REF  REF .
For \(x\in \mathbb {R}^n\)  and \(R>0\) , we see that \(f\in {F}_{pq}^s\big (B\big ((x^{\prime },\frac{x_n}{r}),\max (R,\frac{R}{|r|})\big )\big )\)  implies \(\vartheta ^r f\in {F}_{pq}^s(B(x,R))\) . Therefore
\(&\Vert \vartheta ^rf\Vert _{{F}_{\infty q}^s(\mathbb {R}^n)}\approx \sup _{x\in \mathbb {R}^n,R>0}R^{-\frac{n}{q}}\inf \big \lbrace \Vert \tilde{g}\Vert _{{F}_{qq}^s(\mathbb {R}^n)}:\tilde{g}|_{B(x,R)}=(\vartheta ^rf)|_{B(x,R)}\big \rbrace \\\le &\sup _{x\in \mathbb {R}^n,R>0}R^{-\frac{n}{q}}\inf \big \lbrace \Vert \vartheta ^\frac{1}{r}\tilde{f}\Vert _{{F}_{qq}^s(\mathbb {R}^n)}:\tilde{f}\big |_{B\big ((x^{\prime },\frac{x_n}{r}),\max (R,\frac{R}{|r|})\big )}=f\big |_{B\big ((x^{\prime },\frac{x_n}{r}),\max (R,\frac{R}{|r|})\big )}\big \rbrace \\\lesssim &(|r|^{-M_{q,q}^{n,s}}+|r|^{M_{q,q}^{n,s}})\sup _{x\in \mathbb {R}^n,R>0}R^{-\frac{n}{q}}\Vert  f\Vert _{{F}_{qq}^s\left(B\left((x^{\prime },\frac{x_n}{r}),\max (R,\frac{R}{|r|})\right)\right)}\\\lesssim &(|r|^{-M_{q,q}^{n,s}}+|r|^{M_{q,q}^{n,s}})(1+|r|^{-\frac{n}{q}})\sup _{x\in \mathbb {R}^n,R>0}R^{-\frac{n}{q}}\Vert  f\Vert _{{F}_{qq}^s(B(x,R))}\lesssim (|r|^{-M_{q,q}^{n,s}-\frac{n}{q}}+|r|^{M_{q,q}^{n,s}})\Vert  f\Vert _{{F}_{\infty q}^s(\mathbb {R}^n)}.\) 

Taking \(M_{\infty ,q}^{n,s}:=M_{q,q}^{n,s}+n/q\) , we complete the proof.
We now start to prove Theorem REF  REF . In order to handle Theorem REF  REF  later, we consider a more quantitative version of Theorem REF  REF  that have controls of the operator norms in terms of \((a_j)_j\)  and \((b_j)_j\) .
Theorem 23 (Quantitative common extensions) 
For every \(0<p,q\le \infty \) , \(s\in \mathbb {R}\) , there is a constant \(\widehat{M}=\widehat{M}_{p,q}^{n,s}\) , such that the following holds:
Let \(m_1,m_2\in \mathbb {Z}_{\ge 0}\)  be two non-negative integers such that \(n\max (0,\frac{1}{p}-1,\frac{1}{q}-1)-m_1<s<\frac{1}{p}+m_2\) . Let \(a=(a_j)_{j=-\infty }^\infty \subset \mathbb {R}\)  and \(b=(b_j)_{j=-\infty }^\infty \subset \mathbb {R}_+\)  be two sequences satisfying
\(\sum _{j=-\infty }^\infty a_j(-b_j)^k=1\text{ for }-m_1\le k\le m_2;\quad \sum _{j\in \mathbb {Z}} 2^{\delta |j|}|a_j|(b_j^{-\max (m_1,\widehat{M})}+b_j^{\max (m_2,\widehat{M})})<\infty \text{ for some }\delta >0.\) 
Then \(E=E^{a,b}\)  given by (REF ) has the boundedness \(E:{A}_{pq}^s(\mathbb {R}^n_+)\rightarrow {A}_{pq}^s(\mathbb {R}^n)\)  for \({A}\in \lbrace {B},{F}\rbrace \) . Moreover for the operator norms of \(E\)  we have
\(\Vert Ef\Vert _{{A}_{pq}^s(\mathbb {R}^n)}\le C_{n,p,q,s,\delta ,{\widehat{M}}}\sum _{j\in \mathbb {Z}}2^{\delta |j|}|a_j|\big (b_j^{-\widehat{M}_{p,q}^{n,s}}+b_j^{\widehat{M}_{p,q}^{n,s}}\big )\Vert f\Vert _{{A}_{pq}^s(\mathbb {R}^n_+)}.\) 
Here \(C_{n,p,q,s,\delta ,\widehat{M}}\)  does not depend on the choices of \((a_j)_j\)  and \((b_j)_j\) .

Remark 24 By Proposition REF , the sequences \((a_j)_j\)  and \((b_j)_j\)  can be chosen independently of \(p,q,n,s,m_1,m_2\) . Note that \(\delta \)  in (REF ) is allowed to depend on \(m_1,m_2,\widehat{M}_{pq}^{ns}\)  (cf. the \(\delta _k\)  in (REF ) and Remark REF ).
Therefore, by letting \(m_1,m_2\rightarrow +\infty \)  we obtain the all boundedness properties of \(E\)  in Theorem REF  REF .
Additionally, Theorem REF  also gives the following:

If one take \(m_1=0\)  and let \(m_2\rightarrow +\infty \) , we see that Seeley's extension operator (see Remark REF ) has boundedness \(E:{A}_{pq}^s(\mathbb {R}^n_+)\rightarrow {A}_{pq}^s(\mathbb {R}^n)\)  for \({A}\in \lbrace {B},{F}\rbrace \) , \(p,q\in (0,\infty ]\)  and \(s>n\max (0,\frac{1}{p}-1,\frac{1}{q}-1)\) . It is possible that the range for \(s\)  is not optimal.

If one take \((a_j)_j\)  to be finitely nonzero (depending on the upper bound of \(m_1,m_2\) ), then the qualitative result \(E:{A}_{pq}^s(\mathbb {R}^n_+)\rightarrow {A}_{pq}^s(\mathbb {R}^n)\)  is well-known for common extension operators. See [13]}.


Similar to the proof of Proposition REF , using (REF ) we only need to prove the case \({A}={F}\) .
To make the notation clear, we use \(E_n^{a,b}\)  for the extension operator of \(\mathbb {R}^n_+\)  given in (REF ) associated with sequences \(a=(a_j)_{j=-\infty }^\infty \)  and \(b=(b_j)_{j=-\infty }^\infty \) .
Recall the zero extension operator \(S_nf(x):=\mathbf {1}_{\mathbb {R}^n_+}(x)f(x)\)  from (REF ).
Case A: We consider \(p<\infty \)  and \(s\)  suitably large. We sub-divide the discussion into three parts.
Step A1: The case \(n=1\) , \(0<p<\infty \) , \(0<q\le \infty \)  and \(\frac{1}{p}-1<s<\frac{1}{p}\) , with arbitrary \(m_1,m_2\ge 0\) .
Indeed by [13]} we have boundedness \(S_1:{F}_{pq}^s(\mathbb {R}_+)\rightarrow {F}_{pq}^s(\mathbb {R})\)  at this range. Clearly \(E^{a,b}_1=S_1+\sum _{j=-\infty }^\infty a_j\cdot \vartheta ^{-b_j}\circ S_1\) .
Therefore by Lemma REF  and Proposition REF ,
\(\Vert E^{a,b}_1f\Vert _{{F}_{pq}^s(\mathbb {R})}\lesssim &_{\min (p,q,1),\delta }\Vert S_1f\Vert _{{F}_{pq}^s(\mathbb {R})}+\sum _{j\in \mathbb {Z}}2^{\delta |j|}|a_j|\Vert \vartheta ^{-b_j}S_1f\Vert _{{F}_{pq}^s(\mathbb {R})}\\\lesssim &_{p,q,s}\sum _{j\in \mathbb {Z}}2^{\delta |j|}|a_j|\big (b_j^{-M_{pq}^{1s}}+b_j^{M_{pq}^{1s}}\big )\Vert S_1f\Vert _{{F}_{pq}^s(\mathbb {R})}\lesssim _{p,q,s}\sum _{j\in \mathbb {Z}}2^{\delta |j|}|a_j|\big (b_j^{-M_{pq}^{1s}}+b_j^{M_{pq}^{1s}}\big )\Vert f\Vert _{{F}_{pq}^s(\mathbb {R}_+)}.\) 

We conclude the case by taking \(\widehat{M}_{pq}^{1s}:=M_{pq}^{1s}\) , the constant in Proposition REF .
Step A2: The case \(n=1\) , \(0<p<\infty \) , \(0<q\le \infty \)  and \(s>\frac{1}{p}-1\) , with \(m_1\ge 0\)  and \(m_2>s-\frac{1}{p}\) .
By [10]} \(\Vert f\Vert _{{F}_{pq}^s(U)}\approx _{p,q,s,m_0}\sum _{k=0}^{m_0}\Vert \partial ^kf\Vert _{{F}_{pq}^{s-m_0}(U)}\)  holds for every \(m_0\ge 1\)  and \(U\in \lbrace \mathbb {R}^n,\mathbb {R}^n_+\rbrace \) . Recall from Remark REF  that \(\partial ^k\circ E_1^{a,b}=E_1^{a(-b)^k,b}\circ \partial ^k\)  holds for \(0\le k\le m_2\) .
When \(s-\frac{1}{p}\notin \mathbb {Z}\) , we let \(m_0:=\lceil s-\frac{1}{p}\rceil \) . Therefore by Step A1 (since \(m_2\ge m_0\) ),
\(\Vert E^{a,b}_1f\Vert _{{F}_{pq}^s(\mathbb {R})}\approx &_{p,q,s}\sum _{k=0}^{m_0}\Vert E^{a(-b)^k,b}_1\partial ^kf\Vert _{{F}_{pq}^{s-m_0}}\lesssim _{p,q,s,\delta }\sum _{k=0}^{m_0}\sum _{j\in \mathbb {Z}}2^{\delta |j|}|a_j||\big (b_j^{k-\widehat{M}_{pq}^{1,s-m_0}}+b_j^{k+\widehat{M}_{pq}^{1,s-m_0}}\big )\Vert \partial ^kf\Vert _{{F}_{pq}^{s-m_0}}\\\lesssim &_{m_0}\sum _{j\in \mathbb {Z}}2^{\delta |j|}|a_j||\big (b_j^{-\widehat{M}_{pq}^{1,s-m_0}}+b_j^{m_0+\widehat{M}_{pq}^{1,s-m_0}}\big )\Vert f\Vert _{{F}_{pq}^s(\mathbb {R}_+)}.\) 

In this case we can take \(\widehat{M}_{pq}^{1s}:=\widehat{M}_{pq}^{1,s-m_0}+m_0=\widehat{M}_{pq}^{1,s-\lceil s-\frac{1}{p}\rceil }+\lceil s-\frac{1}{p}\rceil \) .
The case \(s-\frac{1}{p}\in \mathbb {Z}_+\)  follows from the complex interpolation (REF ), where we can take \(\widehat{M}_{pq}^{1s}:=\max (\widehat{M}_{pq}^{1,s-\frac{1}{2}},\widehat{M}_{pq}^{1,s+\frac{1}{2}})\) .
Step A3: The case \(n\ge 1\) , \(0<p<\infty \) , \(0<q\le \infty \)  and \(s>n\max (0,\frac{1}{p}-1,\frac{1}{q}-1)\) , with \(m_1\ge 0\)  and \(m_2>s-\frac{1}{p}\) .
Indeed we have \((E_n^{a,b}f)(x)=E_1^{a,b}(f(x^{\prime },\cdot ))(x_n)\) . Therefore, for any extension \(\tilde{f}\in {F}_{pq}^s(\mathbb {R}^n)\)  of \(f\in {F}_{pq}^s(\mathbb {R}^n_+)\) ,
\(&\Vert E_n^{a,b}f\Vert _{{F}_{pq}^s(\mathbb {R}^n)}\\\approx &_{p,q,s}\Big (\int _{\mathbb {R}^{n-1}}\Vert E_1^{a,b}(f(x^{\prime },\cdot ))\Vert _{{F}_{pq}^s(\mathbb {R})}^pdx^{\prime }\Big )^{1/p}+\Big (\int _{\mathbb {R}}\Vert (E_n^{a,b}f)(\cdot ,x_n)\Vert _{{F}_{pq}^s(\mathbb {R}^{n-1})}^pdx_n\Big )^{1/p}\hspace{-21.68121pt}&(\text{by Proposition~\ref {Prop::Stuff} \ref {Item::Stuff::Fubini}})\\\le &\Vert E_1^{a,b}\Vert _{{F}_{pq}^s}\Big (\int _{\mathbb {R}^{n-1}}\Vert f(x^{\prime },\cdot )\Vert _{{F}_{pq}^s(\mathbb {R}_+)}^pdx^{\prime }\Big )^{1/p}+ \Vert E_1^{a,b}f\Vert _{L^p(\mathbb {R}_{x_n};{F}_{pq}^s(\mathbb {R}^{n-1}_{x^{\prime }}))}\\\lesssim &_{p,q,s}\Vert E_1^{a,b}\Vert _{{F}_{pq}^s}\Vert f\Vert _{L^p(\mathbb {R}^{n-1};{F}_{pq}^s(\mathbb {R}_+))}+\Vert E_1^{a,b}\Vert _{L^p({F}_{pq}^s)}\Vert f\Vert _{L^p(\mathbb {R}_+;{F}_{pq}^s(\mathbb {R}^{n-1}))}\\\le &_{p,q,s,\delta }\sum _{j\in \mathbb {Z}}2^{\delta |j|}|a_j|\big (b_j^{-\widehat{M}_{pq}^{1s}}+b_j^{\widehat{M}_{pq}^{1s}}\big )\Vert \tilde{f}\Vert _{L^p(\mathbb {R}^{n-1};{F}_{pq}^s)}+\sum _{j\in \mathbb {Z}}2^{\delta |j|}|a_j|b_j^{-\frac{1}{p}}\Vert \tilde{f}\Vert _{L^p(\mathbb {R};{F}_{pq}^s)}&(\text{by \xmlelt {hyperlink}{\XMLaddatt {href}{HTag::PfQStep2}Step A2} and Remark~\ref {Rmk::PfSobBdd::VectValuedTmp}})\\\lesssim &_{p,q,s}\sum _{j\in \mathbb {Z}}2^{\delta |j|}|a_j|\big (b_j^{-\widehat{M}_{pq}^{1s}-\frac{1}{p}}+b_j^{\widehat{M}_{pq}^{1s}}\big )\Vert \tilde{f}\Vert _{{F}_{pq}^s(\mathbb {R}^n)}&(\text{by Proposition~\ref {Prop::Stuff} \ref {Item::Stuff::Fubini}}).\) 
Taking the infimum of the extensions \(\tilde{f}\)  over \(f\) , we get the claim with \(\widehat{M}_{pq}^{ns}:=\widehat{M}_{pq}^{1s}+1/p\) .
Case B: We consider \(p=\infty \) , i.e. the \({F}_{\infty q}^s\) -cases, for \(s>-1\) .
Step B1: The case \(n\ge 1\) , \(p=\infty \) , \(0<q\le \infty \)  and \(-1<s<0\) , with arbitrary \(m_1,m_2\ge 0\) .
The proof is identical to Step A1, where we use \(E^{a,b}_n=S_n+\sum _{j=-\infty }^\infty a_j\cdot \vartheta ^{-b_j}\circ S_n\)  and \(S_n:{F}_{pq}^s(\mathbb {R}^n_+)\rightarrow {F}_{pq}^s(\mathbb {R}^n)\)  (see [13]}). We get the result with \(\widehat{M}_{\infty q}^{ns}:=M_{\infty q}^{ns}\) , the constant in Proposition REF .
Step B2: The case \(n\ge 1\) , \(p=\infty \) , \(0<q\le \infty \)  and \(s>-1\) , with \(m_1\ge 0\)  and \(m_2>s\) .
It remains to prove the case \(s\ge 0\) . The proof is similar to Step A2, except we use (see [13]}) \(\Vert f\Vert _{{F}_{\infty q}^s(\mathbb {R}^n)}\approx _{p,q,s,m_0}\sum _{|\gamma |\le m_0}\Vert \partial ^\gamma f\Vert _{{F}_{pq}^{s-m_0}(\mathbb {R}^n)}\) . When \(s\notin \mathbb {Z}_+\) , since \(m_2\ge \lceil s\rceil \)  we get from Step B1 that
\(\Vert E_n^{a,b}f\Vert _{{F}_{pq}^s(\mathbb {R}^n)}\lesssim _{q,s,\delta }\sum _{j\in \mathbb {Z}}2^{\delta |j|}|a_j||\big (b_j^{-\widehat{M}_{\infty q}^{n,s-\lceil s\rceil }}+b_j^{\lceil s\rceil +\widehat{M}_{\infty q}^{n,s-\lceil s\rceil }}\big )\Vert \tilde{f}\Vert _{{F}_{\infty q}^s(\mathbb {R}^n)},\quad \text{ whenever }\tilde{f}|_{\mathbb {R}^n_+}=f.\) 
Taking infimum of the extensions \(\tilde{f}\)  over \(f\)  we get the claim with \(\widehat{M}_{\infty q}^{ns}:=\widehat{M}_{\infty q}^{n,s-\lceil s\rceil }+\lceil s\rceil \) .
The case \(s\in \mathbb {Z}_+\)  follows from the complex interpolation (REF ), where we can take \(\widehat{M}_{\infty q}^{ns}:=\max (\widehat{M}_{\infty q}^{n,s-\frac{1}{2}},\widehat{M}_{\infty q}^{n,s+\frac{1}{2}})\) .
Final step: The general case: \(n\ge 1\) , \(0<p,q\le \infty \)  and \(s\in \mathbb {R}\) , with \(m_1>n\max (0,\frac{1}{p}-1,\frac{1}{q}-1)-s\)  and \(m_2>s-\frac{1}{p}\) . Here we can take \(m_1:=\lfloor n\max (0,\frac{1}{p}-1,\frac{1}{q}-1)-s\rfloor +1\) .
By Remark REF  again we have \(E_n^{a,b}\circ \partial ^\gamma =\partial ^\gamma \circ E_n^{a(-b)^{-\gamma _n},b}\)  for all \(|\gamma |\le m_1\) , where \(a(-b)^{-\gamma _n}=(a_j(-b_j)^{-\gamma _n})_{j=-\infty }^\infty \) . Therefore when \(f\in {F}_{pq}^s(\mathbb {R}^n_+)\)  and \(g_\gamma \in {F}_{pq}^{s+m_1}(\mathbb {R}^n_+)\)  satisfy \(f=\sum _{|\gamma |\le m_1}\partial ^\gamma g_\gamma \) , by Steps A3 and B2 we have
\(\begin{aligned}&\Vert E_n^{a,b}f\Vert _{{F}_{pq}^s(\mathbb {R}^n)}=\Big \Vert \sum _{|\gamma |\le m_1}\partial ^\gamma \circ E_n^{a(-b)^{-\gamma _n},b}g_\gamma \Big \Vert _{{F}_{pq}^s(\mathbb {R}^n)}\\\lesssim &_{\min (p,q,1),m_1}\sum _{|\gamma |\le m_1}\Vert \partial ^\gamma \circ E_n^{a(-b)^{-\gamma _n},b}g_\gamma \Vert _{{F}_{pq}^s(\mathbb {R}^n)}\\\lesssim &_{p,q,s,m_1}\sum _{|\gamma |\le m_1}\Vert E_n^{a(-b)^{-\gamma _n},b}g_\gamma \Vert _{{F}_{pq}^{s+m_1}(\mathbb {R}^n)}&(\text{see \cite [(2.3.8/3)]{TriebelTheoryOfFunctionSpacesI}})\\\lesssim &_{p,q,s,m_1,\delta }\sum _{|\gamma |\le m_1}\sum _{j\in \mathbb {Z}}2^{\delta |j|}|a_j|\big (b_j^{-\widehat{M}_{pq}^{n,s+m_1}-\gamma _n}+b_j^{\widehat{M}_{pq}^{n,s+m_1}-\gamma _n}\big )\Vert g_\gamma \Vert _{{F}_{pq}^{s+m_1}(\mathbb {R}^n)}&(\text{by Steps \xmlelt {hyperlink}{\XMLaddatt {href}{HTag::PfQStep3}A3} and \xmlelt {hyperlink}{\XMLaddatt {href}{HTag::PfQStep3^{\prime }}B2}}).\end{aligned}\) 
Taking the infimum over the decompositions \(f=\sum _{|\alpha |\le m}\partial ^\alpha g_\alpha \) , by Proposition REF  REF  we obtain the claim with \(\widehat{M}_{pq}^{ns}:=\widehat{M}_{pq}^{n,s+m_1}+m_1=\widehat{M}_{pq}^{n,s+\lfloor n\max (0,\frac{1}{p}-1,\frac{1}{q}-1)-s\rfloor +1}+\lfloor n\max (0,\frac{1}{p}-1,\frac{1}{q}-1)-s\rfloor +1\) . This completes the proof of \({A}={F}\) . The case \({A}={B}\)  is done by real interpolations, as discussed in the beginning of the proof.
We now prove Theorem REF  REF , the boundedness on \({B}_{pq}^{s\tau }\) , \({F}_{pq}^{s\tau }\)  and \({N}_{pq}^{s\tau }\) , using Theorem REF .
In the following, for \(m\in \mathbb {Z}_+\)  and \(r>0\) , we define a sequence \(A^{mr}=(A^{mr}_j)_{j=0}^{2m}\)  to be the unique sequence such that \(\sum _{j=0}^{2m}A^{mr}_j(-2^{-j}r)^k=1\)  for integers \(-m\le k\le m\) . Recall from Remark REF  we have
\(A^{mr}_j=(-r2^{-j})^m\prod _{0\le k\le 2m;k\ne j}\frac{r2^{-k}+1}{r2^{-k}-r2^{-j}}=(-1)^m\frac{2^{-jm}}{r^m}\prod _{0\le k\le 2m;k\ne j}\frac{r2^{-k}+1}{2^{-k}-2^{-j}},\quad 0\le j\le 2m.\) 
We see that \(|A^{mr}_j|\lesssim _m r^m+r^{-m}\)  for \(0\le j\le 2m\) , where the implicit constant does not depend on \(r>0\) .
We define
\(\mathbf {E}^{m,r}f(x^{\prime },x_n)=\mathbf {E}^{m,r}_nf(x^{\prime },x_n):={\left\lbrace \begin{array}{ll}\sum _{j=0}^{2m}A^{mr}_jf(x^{\prime },-r2^{-j}x_n)&x_n<0\\f(x)&x_n>0\end{array}\right.}.\) 
By Theorem REF , for every \(0<p,q\le \infty \) , \(s\in \mathbb {R}\)  and \(m>\max (s-\frac{1}{p},-s,n(\frac{1}{p}-1)-s,n(\frac{1}{q}-1)-s)\) , there is a \(C=C_{n,p,q,s,m}>0\) , such that \(\mathbf {E}^{m,r}:{A}_{pq}^s(\mathbb {R}^n_+)\rightarrow {A}_{pq}^s(\mathbb {R}^n)\)  is bounded for \({A}\in \lbrace {B},{F}\rbrace \)  and for all \(r>0\) , with
\(\Vert \mathbf {E}^{m,r}\Vert _{{A}_{pq}^s(\mathbb {R}^n_+)\rightarrow {A}_{pq}^s(\mathbb {R}^n)}\le C_{n,p,q,s,m}(r^{-m\cdot \widehat{M}_{pq}^{ns}}+r^{m\cdot \widehat{M}_{pq}^{ns}}),\quad \forall \, r>0.\) 
Here \(\widehat{M}_{pq}^{ns}>0\)  is the corresponding constant in Theorem REF .
We only need to prove the case \({A}\in \lbrace {B},{F}\rbrace \) . Indeed, when \(\tau <\frac{1}{p}\)  (hence \(p<\infty \) ), the boundedness of \({N}_{pq}^{s\tau }\)  can be deduced from real interpolations (). When \(\tau =\frac{1}{p}\)  we have coincidence \({N}_{pq}^{s\frac{1}{p}}={B}_{\infty q}^s\)  (see (REF )), which is treated in Theorem REF  REF .
We begin with the case \(s<0\) .
Recall from Proposition REF  REF , \(\Vert \tilde{f}\Vert _{{A}_{pq}^{s\tau }(\mathbb {R}^n)}\approx _{n,p,q,s,\tau }\sup _{x\in \mathbb {R}^n;R>0}R^{-n\tau }\Vert \tilde{f}\Vert _{{A}_{pq}^s(B(x,R))}\)  for \({A}\in \lbrace {B},{F}\rbrace \)  (\(p<\infty \)  if \({A}={F}\) ) under the assumption \(s<0\) .
Clearly if \(x_n\ge R\) , then \(\Vert Ef\Vert _{{A}_{pq}^s(B(x,R))}=\Vert f\Vert _{{A}_{pq}^s(B(x,R))}\) .
If \(x_n\le -R\) , then \((E^{a,b}f)|_{B(x,R)}=\sum _{j=-\infty }^\infty a_j\cdot \vartheta ^{-b_j}f\) . Therefore, for every extension \(\tilde{f}\in {A}_{pq}^{s\tau }(\mathbb {R}^n)\)  of \(f\) ,
\(\begin{aligned}&R^{-n\tau }\Vert E^{a,b}f\Vert _{{A}_{pq}^s(B(x,R))}\lesssim _{p,q,\delta }R^{-n\tau }\sum _{j=-\infty }^\infty 2^{\delta |j|} |a_j|\Vert \vartheta ^{-b_j}f\Vert _{{A}_{pq}^s(B(x,R))}&(\text{by Lemma~\ref {Lem::TriSum}})\\\le &R^{-n\tau }\sum _{j=-\infty }^\infty 2^{\delta |j|} |a_j|(b_j^{-M_{pq}^{ns}}+b_j^{M_{pq}^{ns}})\Vert f\Vert _{{A}_{pq}^s\left(B\left((x^{\prime },-\frac{x_n}{b_j}),\max (R,\frac{R}{b_j})\right)\cap \mathbb {R}^ n_+\right)}&(\text{by Proposition~\ref {Prop::Dila}})\\\le &\sum _{j=-\infty }^\infty 2^{\delta |j|} |a_j|(b_j^{-M_{pq}^{ns}}+b_j^{M_{pq}^{ns}})\max (R,\tfrac{R}{b_j})^{-n\tau }\sup _{y\in \mathbb {R}^n}\Vert \tilde{f}\Vert _{{A}_{pq}^s\big (B(y,\max (R, R/{b_j}))\big )}\\\lesssim &_{n,p,q,s,a,b}\sup _{y\in \mathbb {R}^n;\rho >0}\rho ^{-n\tau }\Vert \tilde{f}\Vert _{{A}_{pq}^s(B(y,\rho ))}\approx _{n,p,q,s,\tau } \Vert \tilde{f}\Vert _{{A}_{pq}^{s\tau }(\mathbb {R}^n)}&(\text{by Proposition~\ref {Prop::Stuff} \ref {Item::Stuff::FInftyQ}}).\end{aligned}\) 
Here \(M_{pq}^{ns}\)  is the constant in Proposition REF . Taking the infimum of the extension \(\tilde{f}\)  over \(f\)  we see that
\(R^{-n\tau }\Vert Ef\Vert _{{A}_{pq}^s(B(x,R))}\lesssim _{n,p,q,s,\tau ,a,b}\Vert f\Vert _{{A}_{pq}^{s\tau }(\mathbb {R}^n_+)}\text{ uniformly for }R>0\text{ and }x_n\le -R,\text{ and hence for all }|x_n|\ge R.\) 
The difficult part is to prove the estimate for \(|x_n|\le R\) . In this case \(B(x,R)\subset B((x^{\prime },0),2R)\) , thus \(R^{-n\tau }\Vert Ef\Vert _{{A}_{pq}^s(B(x,R))}\le 2^{n\tau }(2R)^{-n\tau }\Vert Ef\Vert _{{A}_{pq}^s(B((x^{\prime },0),2R))}\) . Since \(E\)  is translation invariant in \(x^{\prime }\) , we can assume \(x^{\prime }=0\)  without loss of generality.
To summarized the discussion above, to prove the boundedness \(E:{A}_{pq}^{s\tau }(\mathbb {R}^n_+)\rightarrow {A}_{pq}^{s\tau }(\mathbb {R}^n)\)  for \(0<p,q\le \infty \) , \(s<0\)  and \(0\le \tau \le \frac{1}{p}\)  (\(p<\infty \)  if \({A}={F}\) ), it remains to show the existence of \(C=C_{n,p,q,s,\tau ,a,b}>0\)  such that
\(R^{-n\tau }\Vert Ef\Vert _{{A}_{pq}^s(B(0,R))}\le C\sup _{\rho >0}\rho ^{-n\tau }\Vert \tilde{f}\Vert _{{A}_{pq}^s(B(0,\rho ))},\quad \forall \, R>0,\quad \forall \, \tilde{f}\in {A}_{pq}^s(\mathbb {R}^n)\text{ such that }\tilde{f}|_{\mathbb {R}^n_+}=f.\) 
In the following we let
\(B_+(0,R):=B(0,R)\cap \mathbb {R}^n_+;\quad J_+:=\lbrace j\in \mathbb {Z}:b_j>1\rbrace ,\quad J_-:=\lbrace j\in \mathbb {Z}:b_j\le 1\rbrace ,\quad \text{thus}\quad \mathbb {Z}=J_+\amalg J_-.\) 
We fix a \(R>0\)  temporally.
For each \(j\in J_+\) , take an extension \(\tilde{f}_j=\tilde{f}_j^R\in {A}_{pq}^s(\mathbb {R}^n)\)  of \(f|_{B_+(0,b_jR)}\)  such that \(\Vert \tilde{f}_j\Vert _{{A}_{pq}^s(\mathbb {R}^n)}\le 2\Vert f\Vert _{{A}_{pq}^s(B_+(0,b_jR))}\) . We let \(\tilde{f}_-=\tilde{f}_-^R\in {A}_{pq}^s(\mathbb {R}^n)\)  be an extension of \(f|_{B_+(0,R)}\)  such that \(\Vert \tilde{f}_-\Vert _{{A}_{pq}^s(\mathbb {R}^n)}\le 2\Vert f\Vert _{{A}_{pq}^s(B_+(0,R))}\) .
Clearly we have the following extension of \(Ef|_{B(0,R)\cap \mathbb {R}^n_-}\) :
\(\tilde{f}_{<0}=\tilde{f}_{<0}^R:=\sum _{j\in J_+} a_j\cdot \vartheta ^{-b_j}\tilde{f}_j+\sum _{j\in J_-}a_j\cdot \vartheta ^{-b_j}\tilde{f}_-.\) 
However \(\tilde{f}_{<0}\)  is not the extension of \(f|_{B_+(0,R)}=Ef|_{B_+(0,R)}\) . To modify it on the domain \(B_+(0,R)\)  we consider
\(\tilde{f}_{>0}=\tilde{f}_{>0}^R:=\tilde{f}_-+\sum _{j\in J_+}\vartheta ^{-1}\circ \mathbf {E}^{m,\frac{1}{b_j}}\big [\vartheta ^{b_j}(\tilde{f}_j-\tilde{f}_-)\big ]=\tilde{f}_-+\sum _{j\in J_+}\sum _{k=0}^{2m}A^{m,\frac{1}{b_j}}_k\cdot (\vartheta ^{2^{-k}}\tilde{f}_j-\vartheta ^{2^{-k}}\tilde{f}_-)\quad \text{in }\mathbb {R}^n_+,\) 
where we have chosen \(m>\max (s-\frac{1}{p},-s,n(\frac{1}{p}-1)-s,n(\frac{1}{q}-1)-s)\) , and \(\mathbf {E}^{m,1/b_j}\)  is given by (REF ).
When \(j\in J_+\)  (i.e \(b_j>1\) ) we have \((\tilde{f}_j-\tilde{f}_-)|_{B(0,R)}=0\) . Thus in this case \(\mathbf {E}^{m,\frac{1}{b_j}}[\vartheta ^{-b_j}(\tilde{f}_j-\tilde{f}_-)]\big |_{B(0,R)}\equiv 0\)  (since \(2^{-k}/b_j<1\)  for \(j\in J_+\)  and \(k\ge 0\) ). We conclude that \(\tilde{f}_{>0}|_{B_+(0,R)}=\tilde{f}_-|_{B_+(0,R)}=f|_{B_+(0,R)}\) .
On the other hand we can rewrite \(\tilde{f}_{<0}\)  as
\(\tilde{f}_{<0}=\sum _{j=-\infty }^\infty a_j\cdot \vartheta ^{-b_j}\tilde{f}_-+\sum _{j\in J_+}a_j\cdot \vartheta ^{-b_j}(\tilde{f}_j-\tilde{f}_-)=E^{a,b}\tilde{f}_-+\sum _{j\in J_+}a_j\cdot \vartheta ^{-1}\circ \vartheta ^{b_j}(\tilde{f}_j-\tilde{f}_-)\quad \text{on }\mathbb {R}^n_-.\) 
Here for global defined function \(\tilde{g}\) , we use the notation \(E\tilde{g}\)  for \(E[(\tilde{g}|_{\mathbb {R}^n_+})]\) .
Therefore, the function \(\tilde{F}=\tilde{F}^R:=\big (\tilde{f}_{>0}|_{\mathbb {R}^n_+}\big )\cup \big (\tilde{f}_{<0}|_{\mathbb {R}^n_-}\big )\)  is an extension of \(Ef|_{B(0,R)}\)  and has the expression
\(\tilde{F}=E^{a,b}\big [\tilde{f}_-\big ]+\sum _{j\in J_+}a_j\cdot \vartheta ^{-1}\circ \mathbf {E}^{m,\frac{1}{b_j}}\big [\vartheta ^{b_j}(\tilde{f}_j-\tilde{f}_-)\big ]\in {S}^{\prime }(\mathbb {R}^n).\) 
We see that
\(\Vert \tilde{F}\Vert _{{A}_{pq}^s(\mathbb {R}^n)}\lesssim &_{p,q,\delta }\Vert E^{a,b}\tilde{f}_-\Vert _{{A}_{pq}^s(\mathbb {R}^n)}+\sum _{j\in J_+}2^{\delta |j|}|a_j|\Vert \mathbf {E}^{m,\frac{1}{b_j}}\Vert _{{A}_{pq}^s}\Vert \vartheta ^{b_j}(\tilde{f}_j-\tilde{f}_-)\Vert _{{A}_{pq}^s(\mathbb {R}^n)}\hspace{-14.45377pt}&(\text{by Lemma~\ref {Lem::TriSum}})\\\lesssim &_{n,p,q,s,m}C^{\prime }_{\delta , a, b,\widehat{M}}\Vert \tilde{f}_-\Vert _{{A}_{pq}^s(\mathbb {R}^n_+)}+\sum _{j\in J_+}2^{\delta |j|}|a_j|b_j^{m\cdot \widehat{M}_{pq}^{ns}}\Vert \vartheta ^{b_j}(\tilde{f}_j-\tilde{f}_-)\Vert _{{A}_{pq}^s}\hspace{-3.61371pt}&(\text{by Theorem~\ref {Thm::QntBFBdd} and (\ref {Eqn::OpNormEb})})\\\lesssim &_{n,p,q,s,a,b,\delta ,\widehat{M}}\Vert \tilde{f}_-\Vert _{{A}_{pq}^s(\mathbb {R}^n)}+\sum _{j\in J_+}2^{\delta |j|}|a_j|b_j^{m\cdot \widehat{M}_{pq}^{ns}}b_j^{M_{pq}^{ns}}\Vert \tilde{f}_j-\tilde{f}_-\Vert _{{A}_{pq}^s(\mathbb {R}^n)}&(\text{by Proposition~\ref {Prop::Dila}})\\\lesssim &_{\min (1,p,q)}\Vert f\Vert _{{A}_{pq}^s(B_+(0,R))}+\sum _{j\in J_+}2^{\delta |j|}|a_j|b_j^{m\cdot \widehat{M}_{pq}^{ns}+M_{pq}^{ns}}\big (\Vert f\Vert _{{A}_{pq}^s(B_+(0,R))}+\Vert f\Vert _{{A}_{pq}^s(B_+(0,b_jR))}\big ).\hspace{-144.54pt}\) 

Here \(C^{\prime }=C^{\prime }_{\delta , a, b,\widehat{M}}=\sum _{j\in \mathbb {Z}}2^{\delta |j|}|a_j|\big (b_j^{\widehat{M}_{pq}^{ns}}+b_j^{-\widehat{M}_{pq}^{ns}}\big )\) .
Notice that all implied constants in the above computation do not depend on the \(R>0\) .
Therefore for general \(R>0\) , applying (REF ) again with a small \(\delta \) , we see that for every extension \(\tilde{f}\)  of \(f\) ,
\(&R^{-n\tau }\Vert Ef\Vert _{{A}_{pq}^s(B(0,R))}\le R^{-n\tau }\Vert \tilde{F}^R\Vert _{{A}_{pq}^s(\mathbb {R}^n)}\\\lesssim &_{p,q,s,m,\delta }C^{\prime }_{\delta a b\widehat{M}}\cdot R^{-n\tau }\Vert f\Vert _{{A}_{pq}^s(B_+(0,R))}+\sum _{j\in J_+}2^{\delta |j|}|a_j|b_j^{C^{\prime \prime }_{npqs\tau m}}(R^{-n\tau }\Vert f\Vert _{{A}_{pq}^s(B_+(0,R))}+(b_jR)^{-n\tau }\Vert f\Vert _{{A}_{pq}^s(B_+(0,b_jR))})\\\lesssim &_{a,b,n,\tau ,m,\widehat{M},M,\delta }\sup _{\rho \in \lbrace 1,b_j:b_j>1\rbrace }\rho ^{-n\tau }\Vert \tilde{f}\Vert _{{A}_{pq}^s(B(0,\rho ))}\lesssim _{n,p,q,s,\tau }\Vert \tilde{f}\Vert _{{A}_{pq}^{s\tau }(\mathbb {R}^n)}.\) 

Here \(C^{\prime \prime }=C^{\prime \prime }_{n,p,q,s,\tau ,m}:=m\cdot \widehat{M}_{pq}^{ns}+M_{pq}^{ns}+n\tau \)  and we let \(\delta >0\)  be such that \(\sum _{j\in \mathbb {Z}}2^{\delta |j|}|a_j|b_j^{\lceil C^{\prime \prime }\rceil }<\infty \) . Taking the infinmum over the extension \(\tilde{f}\) , we obtain (REF ) and conclude the proof of the case \(s<0\) .
For \(s\ge 0\) , we use the following equivalent norm (see [31]}):
\(\Vert \tilde{f}\Vert _{{A}_{pq}^{s\tau }(\mathbb {R}^n)}\approx _{n,p,q,s,\tau ,m}\sum _{|\gamma |\le m}\Vert \partial ^\gamma \tilde{f}\Vert _{{A}_{pq}^{s-m,\tau }(\mathbb {R}^n)},\quad \forall \, m\ge 1.\) 
Using Lemma REF  REF , and taking \(m\ge s+1\)  we have
\(\Vert E^{a,b}f\Vert _{{A}_{pq}^{s\tau }(\mathbb {R}^n)}\approx \sum _{|\gamma |\le m}\Vert E^{a(-b)^{\gamma _n},b}\partial ^\gamma f\Vert _{{A}_{pq}^{s-m,\tau }(\mathbb {R}^n)}\lesssim \sum _{|\gamma |\le m}\Vert \partial ^\gamma f\Vert _{{A}_{pq}^{s-m,\tau }(\mathbb {R}^n_+)}\lesssim \Vert f\Vert _{{A}_{pq}^{s\tau }(\mathbb {R}^n_+)}.\) 
This completes the proof of \(E:{A}_{pq}^{s\tau }(\mathbb {R}^n_+)\rightarrow {A}_{pq}^{s\tau }(\mathbb {R}^n)\)  for \({A}\in \lbrace {B},{F}\rbrace \)  and all \(0<p,q\le \infty \) , \(s\in \mathbb {R}\)  and \(0\le \tau \le \frac{1}{p}\)  (\(p<\infty \)  if \({A}={F}\) ). The case \({A}={N}\)  follows from real interpolations, as mentioned in the beginning of the proof.
Remark 25 By keeping track of the constants, and possibly enlarging the constant \(\widehat{M}=\widehat{M}_{p,q}^{n,s}\)  in Theorem REF , it is likely to prove that (REF ) is also true if we replace \({A}_{pq}^s\)  by \({A}_{pq}^{s\tau }\)  for every \(0\le \tau \le \frac{1}{p}\)  and \({A}\in \lbrace {B},{F},{N}\rbrace \)  (\(p<\infty \)  for \({F}\) -cases).


The Analogy on Bounded Smooth Domains
On bounded smooth domains the analogy of Theorem REF  is the following:
Theorem 26 (Universal Seeley's extension on smooth domains) 
Let \(\Omega \subset \mathbb {R}^n\)  be a bounded smooth domain, and let \(\mathcal {U}\subset \mathbb {R}^n\)  be a bounded neighborhood of \(\overline{\Omega }\) . Then there exists an extension operator \(\mathcal {E}:{S}^{\prime }(\Omega )\rightarrow {E}^{\prime }(\mathcal {U})\)  that maps extendable distributions in \(\Omega \)  to compactly supported distributions in \(\mathcal {U}\) , such that

(One-parameter dependence) There exists a smooth vector field \(X\in C^\infty _c(\mathcal {U};\mathbb {R}^n)\) , such that for every \(x\in \mathcal {U}\backslash \overline{\Omega }\)  and \(f\in C^0(\overline{\Omega })\) , the value \(\mathcal {E}f(x)\)  depends only on the value of \(f\)  on the (unique) integral curve for \(X\)  passing through \(x\)  that contains in \(\Omega \) , which is, the set
\(\lbrace \gamma ^x(t):t\in \mathbb {R},\, \gamma ^x(t)\in \Omega \rbrace \subset \mathcal {U},\text{ where }\dot{\gamma }^x(t)=X(\gamma ^x(t))\ \forall t\in \mathbb {R}\text{ and }\gamma ^x(0)=x.\) 

(Universal boundedeness) \(\mathcal {E}:W^{k,p}(\Omega )\rightarrow W^{k,p}_c(\mathcal {U})\)  and \(\mathcal {E}:C^{k,s}(\Omega )\rightarrow C^{k,s}_c(\mathcal {U})\)  are defined and bounded for all \(k\in \mathbb {Z}\) , \(0<p\le \infty \)  and \(0<s<1\) . In particular \(\mathcal {E}:C^k(\Omega )\rightarrow C^k_c(\mathcal {U})\)  for \(k\ge 0\) .
\(\mathcal {E}:{A}_{pq}^{s\tau }(\Omega )\rightarrow {A}_{pq}^{s\tau }(\mathcal {U})\)  is bounded for \({A}\in \lbrace {B},{F},{N}\rbrace \) , \(p,q\in (0,\infty ]\) , \(s\in \mathbb {R}\)  and \(\tau \in [0,\frac{1}{p}]\)  (\(p<\infty \)  for \({F}\) -cases). In particular \(\mathcal {E}:{A}_{pq}^s(\Omega )\rightarrow {A}_{pq}^s(\mathcal {U})\)  is bounded for \({A}\in \lbrace {B},{F}\rbrace \) , \(0<p,q\le \infty \)  and \(s\in \mathbb {R}\) .


Note that REF  is false if one use the usual patching construction, for example [10]}.
In the proof we need the standard partition of unity argument.
Lemma 27  Let \(\Phi :\mathbb {R}^n\rightarrow \mathbb {R}^n\)  be a bounded diffeomorphism, i.e. \(\Phi \)  is bijective and
\(\sup _{x\in \mathbb {R}^n}|\partial ^\alpha \Phi (x)|+|\partial ^\alpha (\Phi ^{-1})(x)|<\infty ,\quad \forall \,\alpha \in \mathbb {N}_0^n\backslash \lbrace 0\rbrace .\) 
Let \(\chi \in C_c^\infty (\mathbb {R}^n)\)  be a bounded smooth function. Define a linear map \(Tf:=(\chi \cdot f)\circ \Phi \) .

 \(T:W^{k,p}(\mathbb {R}^n)\rightarrow W^{k,p}(\mathbb {R}^n)\) , \(T:C^{k,s}(\mathbb {R}^n)\rightarrow C^{k,s}(\mathbb {R}^n)\)  are both defined and bounded for all \(k\in \mathbb {Z}\) , \(0<p\le \infty \)  and \(0<s<1\) .
In particular \(T:C^k(\mathbb {R}^n)\rightarrow C^k(\mathbb {R}^n)\)  for \(k\in \mathbb {N}_0\) .

 \(T:{A}_{pq}^{s\tau }(\mathbb {R}^n)\rightarrow {A}_{pq}^{s\tau }(\mathbb {R}^n)\)  for \({A}\in \lbrace {B},{F},{N}\rbrace \) , \(p,q\in (0,\infty ]\) , \(s\in \mathbb {R}\)  and \(\tau \in [0,\frac{1}{p}]\)  (\(p<\infty \)  if \({A}={F}\) ).
In particular \(T:{A}_{pq}^s(\mathbb {R}^n)\rightarrow {A}_{pq}^s(\mathbb {R}^n)\)  is bounded for \({A}\in \lbrace {B},{F}\rbrace \) , for \(p,q\in (0,\infty ]\)  and \(s\in \mathbb {R}\) .


The result REF  follows from using chain rules, product rules and the formula of change of variables:
\(\int _{\mathbb {R}^n}|(\chi f)(\Phi (x))|^pdx=\int _{\mathbb {R}^n}|\chi (y) f(y)|^p|\det \nabla (\Phi ^{-1})(y)|dy,\quad 0<p<\infty ,\quad f\in L^p_\mathrm {loc}(\mathbb {R}^n).\) 
Notice that \(|\det \nabla (\Phi ^{-1})|\)  is a bounded smooth function, and all nonzero derivatives of \(\Phi \)  and \(\chi \)  are bounded. We leave the details to readers.
For REF , see for example [33]} and [18]}.
In the following, we let \({X}\)  be the function class that belongs to the following:
\(\lbrace W^{k,p},C^{k,s}:k\in \mathbb {Z},\, p\in (0,\infty ],\,s\in (0,1)\rbrace \cup \lbrace {B}_{pq}^s,{F}_{pq}^s,{B}_{pq}^{s\tau },{N}_{pq}^{s\tau }:p,q\in (0,\infty ],\, s\in \mathbb {R},\,\tau \in [0,\tfrac{1}{p}]\rbrace \\\cup \lbrace C^k:k\in \mathbb {N}_0\rbrace \cup \lbrace {F}_{pq}^{s\tau }:p\in (0,\infty ),\,q\in (0,\infty ],\, s\in \mathbb {R},\,\tau \in [0,\tfrac{1}{p}]\rbrace .\) 
Recall from Theorem REF  that \(E^{a,b}:{X}(\mathbb {R}^n_+)\rightarrow {X}(\mathbb {R}^n)\)  is bounded for all such \({X}\) .
For a function \(g(\theta ,t)\)  defined on \(\partial \Omega \times \mathbb {R}_+\) , we define its extension \(\mathcal {E}_0g\)  on \(\partial \Omega \times \mathbb {R}\)  by
\(\mathcal {E}_0g(\theta ,t):={\left\lbrace \begin{array}{ll}\sum _{j=-\infty }^\infty a_j\cdot g(\theta ,-b_jt)&t<0;\\g(\theta ,t)&t>0\end{array}\right.},\quad \text{where }(a_j,b_j)_{j=-\infty }^\infty \text{ satisfies }(\ref {Eqn::CondAB}).\) 
Recall from Proposition REF  that such \((a_j,b_j)_j\)  exist.
We construct the following objects:

A tubular neighborhood \(U_0\subset \subset \mathcal {U}\)  of \(\partial \Omega \)  along with a diffeomorphismWe do not say \(\Psi \)  a coordinate chart of \(U_0\)  because its codomain is not a subset of \(\mathbb {R}^n\) . \(\Psi =(\psi ^{\prime },\psi _n):U_0\xrightarrow{}\partial \Omega \times (-1,1)\)  such that \(U_0\cap \Omega =\lbrace \psi _n>0\rbrace \) .

A coordinate cover \(\lbrace \phi _\lambda ^{\prime }:V_\lambda \subseteq \partial \Omega \rightarrow \mathbb {R}^{n-1}\rbrace _{\lambda =1}^N\)  of \(\partial \Omega \) , where \(V_\lambda \)  are open in the submanifold \(\partial \Omega \) , along with the coordinate charts \(\Phi _\lambda (\theta ,t):=(\phi ^{\prime }_\lambda (\theta ),t)\)  defined on \(V_\lambda \times (-1,1)\subset \partial \Omega \times (-1,1)\) .

By choosing \(U_0\)  and each \(V_\lambda \)  small enough, we can ensure that

For every \(1\le \lambda \le N\) , \(\Phi _\lambda \circ \Psi :\Psi ^{-1}(V_\lambda \times (-1,1))\subseteq \mathcal {U}\rightarrow \mathbb {R}^n\)  can extend to a bounded diffeomorphism. In other words, there are \(\Xi _\lambda :\mathbb {R}^n\rightarrow \mathbb {R}^n\)  that satisfies (REF ) and \(\Xi _\lambda |_{\Psi ^{-1}(V_\lambda \times (-1,1))}=\Phi _\lambda \circ \Psi \) , for each \(\lambda \) .

For such \(\Psi \)  and \(\lbrace \phi _\lambda ^{\prime }\rbrace _\lambda \) , we take the following cutoff functions:
\(\chi _0,\chi _1\in C_c^\infty (-1,1)\text{ such that }\chi _0|_{[-\frac{1}{2},\frac{1}{2}]}\equiv 1,\ \chi _1|_{\operatorname{supp}\chi _0}\equiv 1;\quad \text{and}\quad \tilde{\chi }_0:=\chi _0\circ \psi _n,\ \tilde{\chi }_1:=\chi _1\circ \psi _n\in C_c^\infty (U_0).\) 
We define \(\mathcal {E}\)  as
\(\mathcal {E}f:=(1-\tilde{\chi }_0)|_\Omega \cdot f+\tilde{\chi }_1\cdot (\mathcal {E}_0[(\tilde{\chi }_0\cdot f)\circ \Psi ^{-1}]\circ \Psi ).\) 
Clearly \(\mathcal {E}f|_\Omega =(1-\tilde{\chi }_0)\cdot f+\tilde{\chi }_1\cdot \tilde{\chi }_0\cdot f=f\) . Therefore \(\mathcal {E}\)  is an extension operator.
Take \(\tilde{\chi }_2\in C_c^\infty (\mathcal {U})\)  be such that \(\tilde{\chi }_2|_{\operatorname{supp}\tilde{\chi }_1}\equiv 1\) . Immediately we see that the vector field \(X:=\tilde{\chi }_2\cdot (\Psi ^*\frac{\partial }{\partial t})\)  is what we need for REF .
We postpone the proof of \(\mathcal {E}:{S}^{\prime }(\Omega )\rightarrow {E}^{\prime }(\mathcal {U})\)  to the end.
For REF , to prove the boundedness for \(\tilde{\chi }_1(\mathcal {E}_0[(\tilde{\chi }_0f)\circ \Psi ^{-1}]\circ \Psi )\) , we take the following objects:

\(\lbrace \mu _\lambda \in C_c^\infty (V_\lambda )\rbrace _{\lambda =1}^N\)  be a partition of unity of \(\partial \Omega \) , i.e. \(\sum _{\lambda =1}^N\mu _\lambda \equiv 1\)  on \(\partial \Omega \) .

\(\lbrace \rho _\lambda \in C_c^\infty (V_\lambda )\rbrace _{\lambda =1}^N\)  be such that \(\rho _\lambda |_{\operatorname{supp}\mu _\lambda }\equiv 1\) .

For \(1\le \lambda \le N\) , we let
\(\tilde{\mu }_\lambda :=(\mu _\lambda \otimes \chi _0)\circ \Psi =(\mu _\lambda \circ \psi ^{\prime })\cdot (\chi _0\circ \psi _n)\quad \text{and}\quad \tilde{\rho }_\lambda (x):=(\rho _\lambda \otimes \chi _1)\circ \Phi _\lambda ^{-1}(x)=(\rho _\lambda \circ {\phi ^{\prime }_\lambda }^{-1})(x^{\prime })\cdot \chi _1(x_n).\) 

Immediately we see that

\(\tilde{\mu }_\lambda \in C_c^\infty (\Psi ^{-1}(V_\lambda \times (-1,1)))\)  and \(\tilde{\rho }_\lambda \in C_c^\infty (\Phi _\lambda (V_\lambda \times (-1,1)))\) .

When \(\operatorname{supp}g\subseteq (\operatorname{supp}\mu _\lambda )\times [0,\infty )\) , we have \(\operatorname{supp}\mathcal {E}_0g\subseteq (\operatorname{supp}\mu _\lambda )\times \mathbb {R}\) .
In particular \((\mathbf {1}_{\partial \Omega }\otimes \chi _1)\cdot \mathcal {E}_0[(\mu _\lambda \otimes \chi _0)\cdot g]=(\rho _\lambda \otimes \chi _1)\cdot \mathcal {E}_0[(\mu _\lambda \otimes \chi _0)\cdot g]\)  (since \(\rho _\lambda |_{\operatorname{supp}\mu _\lambda }\equiv 1\) ).

When \(\operatorname{supp}g\subseteq V_\lambda \times [0,1)\) , we have \(\mathcal {E}_0[g]|_{V_\lambda \times (-1,1)}=E[g\circ \Phi _\lambda ^{-1}]\circ \Phi _\lambda \) , the right hand side of which is well-defined. Here \(E\)  is the extension operator from (REF ).

Therefore
\(&\tilde{\chi }_1\cdot (\mathcal {E}_0[(\tilde{\chi }_0\cdot f)\circ \Psi ^{-1}]\circ \Psi )=\sum _{\lambda =1}^N\big ((\mathbf {1}_{\partial \Omega }\otimes \chi _1)\cdot \mathcal {E}_0[(\tilde{\mu }_\lambda \cdot f)\circ \Psi ^{-1}]\big )\circ \Psi \\=&\sum _{\lambda =1}^N\left((\rho _\lambda \otimes \chi _1)\cdot \big ( E[(\tilde{\mu }_\lambda \cdot f)\circ \Psi ^{-1}\circ \Phi _\lambda ^{-1}]\circ \Phi _\lambda \big )\right)\circ \Psi =\sum _{\lambda =1}^N\left(\tilde{\rho }_\lambda \cdot E[(\tilde{\mu }_\lambda \cdot f)\circ (\Phi _\lambda \circ \Psi )^{-1}]\right)\circ (\Phi _\lambda \circ \Psi ).\) 

Applying Lemma REF  and Theorem REF  REF  - REF , we have the boundedness of compositions
\({X}(\Omega )\xrightarrow{}{X}(\mathbb {R}^n_+)^N\xrightarrow{}{X}(\mathbb {R}^n)^N\xrightarrow{}{X}_c(U_0)\subset {X}_c(\mathcal {U}).\) 
On the other hand, \(\tilde{\chi }_0=\chi _0\circ \psi _n\in C_c^\infty (U_0)\)  is identical to 1 near \(\partial \Omega \) , therefore \((1-\tilde{\chi }_0)|_\Omega \in C_c^\infty (\Omega )\) . By Lemma REF  \([f\mapsto (1-\tilde{\chi }_0)|_\Omega \cdot f]:{X}(\Omega )\rightarrow {X}_c(\mathcal {U})\)  is bounded. Altogether we conclude that \(\mathcal {E}:{X}(\Omega )\rightarrow {X}(\mathcal {U})\)  is bounded. This completes the proof of REF .
Finally we illustrate the definedness of \(\mathcal {E}:{S}^{\prime }(\Omega )\rightarrow {E}^{\prime }(\mathcal {U})\) . Note that alternatively one can obtain this statement by using duality and prove the boundedness \(\mathcal {E}^*:C_\mathrm {loc}^\infty (\mathcal {U})\rightarrow {S}(\Omega )(=\lbrace g\in C^\infty (\mathbb {R}^n):\operatorname{supp}g\subseteq \overline{\Omega }\rbrace )\) .
For a \(f\in {S}^{\prime }(\Omega )\) , let \(\tilde{f}\)  be an extension of \(f\) . Since \(\Omega \)  is a bounded domain, we can assume \(\tilde{f}\)  to have compact support, otherwise we replace \(\tilde{f}\)  with \(\kappa \tilde{f}\)  where \(\kappa \in C_c^\infty (\mathbb {R}^n)\)  is such that \(\kappa |_{\Omega }\equiv 1\) .
By the structure theorem of distributions (see [35]} for example), there are \(M\ge 0\)  and \(\lbrace g_\alpha \rbrace _{|\alpha |\le M}\subset C_c^0(\mathbb {R}^n)\)  such that \(\tilde{f}=\sum _{|\alpha |\le M}\partial ^\alpha g_\alpha \) . By Proposition REF  REF  this is saying that \(\tilde{f}\in {B}_{\infty \infty }^{-M}(\mathbb {R}^n)\)  for some \(M>0\) , which means \(f\in {B}_{\infty \infty }^{-M}(\Omega )\)  for the same \(M\) .
Now by REF  \(\mathcal {E}f\in {B}_{\infty \infty }^{-M}(\mathcal {U})\subset {D}^{\prime }(\mathcal {U})\)  has compact support, we conclude that \(\mathcal {E}f\in E^{\prime }(\mathcal {U})\) .






